{"id":185033,"date":"2022-06-08T17:30:00","date_gmt":"2022-06-08T12:00:00","guid":{"rendered":"https:\/\/www.aakash.ac.in\/blog\/?post_type=web-story&#038;p=185033"},"modified":"2022-09-03T12:43:26","modified_gmt":"2022-09-03T07:13:26","slug":"maths-rolles-theorem-class-12-ncert-explained-with-examples","status":"publish","type":"web-story","link":"https:\/\/www.aakash.ac.in\/blog\/web-stories\/maths-rolles-theorem-class-12-ncert-explained-with-examples\/","title":{"rendered":"Rolle\u2019s Theorem Class 12 NCERT: Explained With Examples"},"content":{"rendered":"<p><html amp=\"\" lang=\"en\"><head><meta charSet=\"utf-8\"\/><meta name=\"viewport\" content=\"width=device-width,minimum-scale=1,initial-scale=1\"\/><script async=\"\" src=\"https:\/\/cdn.ampproject.org\/v0.js\"><\/script><script async=\"\" src=\"https:\/\/cdn.ampproject.org\/v0\/amp-story-1.0.js\" custom-element=\"amp-story\"><\/script><link 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publisher-logo-src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/08\/Aakash-Icon-Blue-PNG.png\" title=\"Rolle\u2019s Theorem Class 12 NCERT: Explained With Examples\" poster-portrait-src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/cropped-Rolles-theorem-4.jpg\"><amp-story-page id=\"568b5f37-c29f-440f-8679-4637776b9f67\" auto-advance-after=\"9s\"><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\" style=\"background-color:#fff\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:0;top:-9.25926%;width:100%;height:118.51852%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-93d5b085-d6c2-4dd6-9b2f-0a4c3bc02de3\">\n<div style=\"position:absolute;width:100%;height:100%;left:0%;top:0%\" data-leaf-element=\"true\"><amp-img 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style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:0.6349206349206349% 0.6349206349206349% 0.6349206349206349% 0.6349206349206349% \/ 2.197802197802198% 2.197802197802198% 2.197802197802198% 2.197802197802198%\" id=\"el-8d19cf2a-e493-45fb-9f83-a4b9f1bef8dd\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:0.22619047619047575% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.404531em;line-height:1.2;text-align:center;padding:0;color:#000000\"><span><span style=\"font-weight: 700; color: #edf6f6\">Rolle\u2019s Theorem Class 12 NCERT: Explained With Examples<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:10.92233%;top:77.18447%;width:76.69903%;height:1.13269%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-8a4007b8-de83-4611-af8e-86e250f5f078\">\n<div class=\"fill\" style=\"background-color:#c4c4c4;will-change:transform\"><\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:11.65049%;top:95.63107%;width:76.69903%;height:1.13269%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-d7c93611-2195-4d8e-a991-f4b4e256e0ba\">\n<div class=\"fill\" style=\"background-color:#c4c4c4;will-change:transform\"><\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:5.82524%;top:21.19741%;width:88.34951%;height:53.07443%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-54ba6ce5-88f4-4f7d-8d3f-9c36e60d82c4\">\n<div style=\"position:absolute;width:111.21728%;height:219.42054%;left:-5.67719%;top:-44.14112%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-4-1-1.png\" alt=\"Rolle&#x27;s theorem-4-1 (1)\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-4-1-1.png 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-4-1-1-576x1024.png 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-4-1-1-169x300.png 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-4-1-1-150x267.png 150w\" sizes=\"(min-width: 1024px) 40vh, 88vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:11.8932%;top:0;width:14.07767%;height:9.38511%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" 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1)\",\"fill\":\"forwards\"}]<\/script><\/amp-story-animation><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\" style=\"background-color:#0f1628\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:0;top:-9.25926%;width:100%;height:118.51852%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-df36fa65-078b-4e1d-8e73-c9d9f6d2f7ca\">\n<div style=\"position:absolute;width:100%;height:100%;left:0%;top:0%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/24-1.jpg\" alt=\"24\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/24-1.jpg 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/24-1-576x1024.jpg 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/24-1-169x300.jpg 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/24-1-150x267.jpg 150w\" sizes=\"(min-width: 1024px) 45vh, 100vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:11.8932%;top:75.08091%;width:76.45631%;height:18.12298%;opacity:1\">\n<div id=\"anim-513c517d-6601-4f40-a9c4-95432adb9848\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:translate3d(-115.55555%, 0px, 0)\">\n<div id=\"anim-8ddc7cb3-469a-46a7-93be-ac47b76e2433\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:0;--initial-transform:none\">\n<div id=\"anim-d604627c-e462-4637-87e1-426d4d56819d\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:scale(0.15)\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:0.6349206349206349% 0.6349206349206349% 0.6349206349206349% 0.6349206349206349% \/ 1.7857142857142856% 1.7857142857142856% 1.7857142857142856% 1.7857142857142856%\" id=\"el-f132b5e0-e9cf-46e5-8ecb-4865c4fb24d1\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:0.20809523809523808% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.372168em;line-height:1.2;text-align:center;padding:0;color:#000000\"><span><span style=\"color: #edf6f6\">In Class 12, <\/span><span style=\"font-weight: 700; color: #edf6f6\">Rolle&#8217;s Theorem<\/span><span style=\"color: #edf6f6\"> is a variant of the mean value theorem that satisfies certain conditions.<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:0;top:15.21036%;width:98.05825%;height:66.66667%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-dfd1c182-0593-4da5-afc5-73bce20473c0\">\n<div style=\"position:absolute;width:100%;height:174.32577%;left:0%;top:-37.16288%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-5-1.png\" alt=\"Rolle&#x27;s theorem-5-1\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-5-1.png 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-5-1-576x1024.png 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-5-1-169x300.png 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-5-1-150x267.png 150w\" sizes=\"(min-width: 1024px) 44vh, 98vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><\/amp-story-page><amp-story-page id=\"df4d1f91-7fcb-449e-902e-f00e376eac9f\" auto-advance-after=\"9s\"><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-29baf2bd-aa3c-4804-9db4-2d76ccbd46e0\",\"keyframes\":{\"transform\":[\"translate3d(-115.2381%, 0px, 0)\",\"translate3d(0px, 0px, 0)\"]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(0.4, 0.4, 0.0, 1)\",\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-animation 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style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-4b1e743a-2cc3-438b-836e-d05a9361883e\">\n<div style=\"position:absolute;width:100%;height:100%;left:0%;top:0%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/58.jpg\" alt=\"58\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/58.jpg 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/58-576x1024.jpg 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/58-169x300.jpg 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/58-150x267.jpg 150w\" sizes=\"(min-width: 1024px) 45vh, 100vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:11.65049%;top:9.06149%;width:76.45631%;height:18.12298%;opacity:1\">\n<div id=\"anim-29baf2bd-aa3c-4804-9db4-2d76ccbd46e0\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:translate3d(-115.2381%, 0px, 0)\">\n<div id=\"anim-3a6bd4fc-e5b1-4145-b63d-e105f81fdafe\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:0;--initial-transform:none\">\n<div id=\"anim-7c0c0675-5a38-4c51-ab84-8bf007e275da\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:scale(0.15)\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:0.6349206349206349% 0.6349206349206349% 0.6349206349206349% 0.6349206349206349% \/ 1.7857142857142856% 1.7857142857142856% 1.7857142857142856% 1.7857142857142856%\" id=\"el-8c50ba34-8a0e-4688-afa1-32033aca6ab3\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:0.20809523809523808% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.372168em;line-height:1.2;text-align:center;padding:0;color:#000000\"><span><span style=\"color: #edf6f6\">So, let us begin with learning the \u2018Lagrange&#8217;s Mean Value Theorem\u2019 (LMVT).<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:1.21359%;top:18.60841%;width:97.3301%;height:94.82201%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-85080a37-70aa-45a8-95d8-27f628aa732d\">\n<div style=\"position:absolute;width:100%;height:121.65339%;left:0%;top:-10.8267%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-1-1.png\" alt=\"Rolle&#x27;s theorem-1-1\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-1-1.png 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-1-1-576x1024.png 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-1-1-169x300.png 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-1-1-150x267.png 150w\" sizes=\"(min-width: 1024px) 44vh, 97vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><\/amp-story-page><amp-story-page id=\"d930b368-8912-47c9-8ea9-696400a9ff1e\" auto-advance-after=\"9s\"><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-da63d689-1131-4ae6-9ccc-3c37a1da3f10\",\"keyframes\":[{\"offset\":0,\"transform\":\"translate3d(0, -707.778%, 0)\",\"easing\":\"cubic-bezier(.5, 0, 1, 1)\"},{\"offset\":0.29,\"transform\":\"translate3d(0, 0%, 0)\",\"easing\":\"cubic-bezier(0, 0, .5, 1)\"},{\"offset\":0.45,\"transform\":\"translate3d(0, -199.0271736%, 0)\",\"easing\":\"cubic-bezier(.5, 0, 1, 1)\"},{\"offset\":0.61,\"transform\":\"translate3d(0, 0%, 0)\",\"easing\":\"cubic-bezier(0, 0, .5, 1)\"},{\"offset\":0.71,\"transform\":\"translate3d(0, -67.6635768%, 0)\",\"easing\":\"cubic-bezier(.5, 0, 1, 1)\"},{\"offset\":0.8,\"transform\":\"translate3d(0, 0%, 0)\",\"easing\":\"cubic-bezier(0, 0, .5, 1)\"},{\"offset\":0.85,\"transform\":\"translate3d(0, -25.409230200000003%, 0)\",\"easing\":\"cubic-bezier(.5, 0, 1, 1)\"},{\"offset\":0.92,\"transform\":\"translate3d(0, 0%, 0)\",\"easing\":\"cubic-bezier(0, 0, .5, 1)\"},{\"offset\":0.96,\"transform\":\"translate3d(0, -11.0413368%, 0)\",\"easing\":\"cubic-bezier(.5, 0, 1, 1)\"},{\"offset\":1,\"transform\":\"translate3d(0, 0%, 0)\",\"easing\":\"cubic-bezier(0, 0, .5, 1)\"}],\"delay\":0,\"duration\":1300,\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\" style=\"background-color:#1a92cb\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:0;top:-9.25926%;width:100%;height:118.51852%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" 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cta-image=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/08\/Aakash-Icon-Blue-PNG.png\"><a href=\"https:\/\/byjus-aakash.onelink.me\/GieR\/aakashwebstories\" rel=\"\">Download Aakash App<\/a><\/amp-story-page-outlink><\/amp-story-page><amp-story-page id=\"cf8413e8-8638-4f52-ab50-24cfd17449fa\" auto-advance-after=\"9s\"><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-f933481c-a1db-4e6f-a7a3-123a9e585fdd\",\"keyframes\":{\"transform\":[\"translate3d(-115.55555%, 0px, 0)\",\"translate3d(0px, 0px, 0)\"]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(0.4, 0.4, 0.0, 1)\",\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-3ac87db3-39fa-4790-a4f0-6f95bc7b65a8\",\"keyframes\":{\"opacity\":[0,1]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(0.4, 0.4, 0.0, 1)\",\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-d827ba7b-eb99-4e4c-afd8-40d4b94d17cf\",\"keyframes\":{\"transform\":[\"scale(0.15)\",\"scale(1)\"]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(0.4, 0.4, 0.0, 1)\",\"fill\":\"forwards\"}]<\/script><\/amp-story-animation><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\" style=\"background-color:#fbfbf9\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:0;top:-9.25926%;width:100%;height:118.51852%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-fb0206f3-1b7f-4842-9497-54c72b11ec3f\">\n<div style=\"position:absolute;width:100%;height:100%;left:0%;top:0%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/44.jpg\" alt=\"44\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/44.jpg 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/44-576x1024.jpg 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/44-169x300.jpg 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/44-150x267.jpg 150w\" sizes=\"(min-width: 1024px) 45vh, 100vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:11.8932%;top:14.40129%;width:76.45631%;height:21.68285%;opacity:1\">\n<div id=\"anim-f933481c-a1db-4e6f-a7a3-123a9e585fdd\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:translate3d(-115.55555%, 0px, 0)\">\n<div id=\"anim-3ac87db3-39fa-4790-a4f0-6f95bc7b65a8\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:0;--initial-transform:none\">\n<div id=\"anim-d827ba7b-eb99-4e4c-afd8-40d4b94d17cf\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:scale(0.15)\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:0.6349206349206349% 0.6349206349206349% 0.6349206349206349% 0.6349206349206349% \/ 1.4925373134328357% 1.4925373134328357% 1.4925373134328357% 1.4925373134328357%\" id=\"el-231a6f41-5705-4451-8abb-9f1e8187dc5f\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:0.1990476190476187% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.355987em;line-height:1.2;text-align:center;padding:0;color:#000000\"><span><span style=\"color: #edf6f6\">It states that:<\/span><br \/>\n<span style=\"color: #edf6f6\">A function is defined on a closed interval [a,b] if it satisfies the below condition:<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:11.65049%;top:39.32039%;width:80.09709%;height:46.92557%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-97bdca72-1ca7-4c21-a7d3-b9a3e426a129\">\n<div style=\"position:absolute;width:100%;height:202.29884%;left:0%;top:-32.64888%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-3-1-1.png\" alt=\"Rolle&#x27;s theorem-3-1 (1)\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-3-1-1.png 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-3-1-1-576x1024.png 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-3-1-1-169x300.png 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-3-1-1-150x267.png 150w\" sizes=\"(min-width: 1024px) 36vh, 80vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><\/amp-story-page><amp-story-page id=\"43946efa-4942-4387-b717-5a141e0a7118\" auto-advance-after=\"9s\"><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-0a026713-b05c-4f6f-95d3-9be2b933454b\",\"keyframes\":{\"opacity\":[0,1]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-fec0a952-5890-4032-ad1b-74ffc01670d1\",\"keyframes\":{\"transform\":[\"scale(0.3333333333333333)\",\"scale(1)\"]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"forwards\"}]<\/script><\/amp-story-animation><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\" style=\"background-color:#1b2b3a\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:0;top:-9.25926%;width:100%;height:118.51852%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-c656ca66-f1aa-4008-b31a-ab9b3c530b40\">\n<div style=\"position:absolute;width:100%;height:100%;left:0%;top:0%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/43.jpg\" alt=\"43\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/43.jpg 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/43-576x1024.jpg 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/43-169x300.jpg 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/43-150x267.jpg 150w\" sizes=\"(min-width: 1024px) 45vh, 100vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:11.8932%;top:8.25243%;width:76.69903%;height:19.41748%;opacity:1\">\n<div id=\"anim-0a026713-b05c-4f6f-95d3-9be2b933454b\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:0;--initial-transform:none\">\n<div id=\"anim-fec0a952-5890-4032-ad1b-74ffc01670d1\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:scale(0.3333333333333333)\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:0.6329113924050633% 0.6329113924050633% 0.6329113924050633% 0.6329113924050633% \/ 1.6666666666666667% 1.6666666666666667% 1.6666666666666667% 1.6666666666666667%\" id=\"el-ac3d2033-e30c-4985-a14f-73d9cb09da37\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:-0.4977848101265824% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.355987em;line-height:1.4;text-align:center;padding:0;color:#000000\"><span><span style=\"color: #edf6f6\">Note: Lagrange\u2019s mean&nbsp;<\/span><br \/>\n<span style=\"color: #edf6f6\">value theorem is&nbsp;<\/span><br \/>\n<span style=\"color: #edf6f6\">also known as&nbsp;<\/span><br \/>\n<span style=\"color: #ffd900\">\u2018first mean value theorem\u2019.<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:0.97087%;top:20.71197%;width:99.51456%;height:94.82201%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-3f1a446d-6be9-47dc-a1f4-7e11796ef6fb\">\n<div style=\"position:absolute;width:100%;height:124.38376%;left:0%;top:-12.19188%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-4-1-1.png\" alt=\"Rolle&#x27;s theorem-4-1 (1)\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-4-1-1.png 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-4-1-1-576x1024.png 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-4-1-1-169x300.png 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-4-1-1-150x267.png 150w\" sizes=\"(min-width: 1024px) 45vh, 100vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><\/amp-story-page><amp-story-page id=\"b23efef2-26d6-41b1-9148-63ed8991b99e\" auto-advance-after=\"9s\"><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-6abf1064-66fa-416a-b266-99b0ba0a5154\",\"keyframes\":{\"opacity\":[0,1]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-2b5b7489-23a8-423d-acf1-6c337e299646\",\"keyframes\":{\"transform\":[\"scale(0.3333333333333333)\",\"scale(1)\"]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"forwards\"}]<\/script><\/amp-story-animation><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-8460a5ca-79ee-44af-8aee-2fc286222861\",\"keyframes\":{\"opacity\":[0,1]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-f63ebf91-a42c-4970-a7df-758f25e8beed\",\"keyframes\":{\"transform\":[\"scale(0.3333333333333333)\",\"scale(1)\"]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"forwards\"}]<\/script><\/amp-story-animation><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\" style=\"background-color:#253c44\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:0;top:-9.25926%;width:100%;height:118.51852%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-e1ca807d-0573-496a-baaf-02322961a0c0\">\n<div style=\"position:absolute;width:100%;height:100%;left:0%;top:0%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/56.jpg\" alt=\"56\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/56.jpg 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/56-576x1024.jpg 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/56-169x300.jpg 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/56-150x267.jpg 150w\" sizes=\"(min-width: 1024px) 45vh, 100vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:11.65049%;top:4.36893%;width:76.69903%;height:14.23948%;opacity:1\">\n<div id=\"anim-6abf1064-66fa-416a-b266-99b0ba0a5154\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:0;--initial-transform:none\">\n<div id=\"anim-2b5b7489-23a8-423d-acf1-6c337e299646\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:scale(0.3333333333333333)\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:0.6329113924050633% 0.6329113924050633% 0.6329113924050633% 0.6329113924050633% \/ 2.272727272727273% 2.272727272727273% 2.272727272727273% 2.272727272727273%\" id=\"el-1da4d649-875a-44a4-9e58-4d76dd15e723\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:-0.4977848101265824% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.355987em;line-height:1.4;text-align:center;padding:0;color:#000000\"><span><span style=\"color: #edf6f6\">Now, let us see the special case of LMVT, that defines Rolle&#8217;s Theorem.<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:11.65049%;top:75.72816%;width:76.69903%;height:24.27184%;opacity:1\">\n<div id=\"anim-8460a5ca-79ee-44af-8aee-2fc286222861\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:0;--initial-transform:none\">\n<div id=\"anim-f63ebf91-a42c-4970-a7df-758f25e8beed\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:scale(0.3333333333333333)\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:0.6329113924050633% 0.6329113924050633% 0.6329113924050633% 0.6329113924050633% \/ 1.3333333333333335% 1.3333333333333335% 1.3333333333333335% 1.3333333333333335%\" id=\"el-92d7383e-9b56-45f4-b90e-305e960e03c5\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:-0.4977848101265824% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.355987em;line-height:1.4;text-align:center;padding:0;color:#000000\"><span><span style=\"color: #edf6f6\">The first conditions are the same for both LMVT and Rolle\u2019s Theorem, but the third special condition for Rolle\u2019s theorem is:<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:9.95146%;top:23.7055%;width:80.09709%;height:49.35275%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-56038d56-ebb0-4e8f-a119-b3e38c5d6b7a\">\n<div style=\"position:absolute;width:100%;height:192.34974%;left:0%;top:-25.4177%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-6-1.png\" alt=\"Rolle&#x27;s theorem-6-1\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-6-1.png 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-6-1-576x1024.png 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-6-1-169x300.png 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-6-1-150x267.png 150w\" sizes=\"(min-width: 1024px) 36vh, 80vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><\/amp-story-page><amp-story-page id=\"bfeba4cf-8e0d-47bb-97e6-671bcfe35fd2\" auto-advance-after=\"9s\"><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-7f909712-243b-4be9-970f-737b178ce317\",\"keyframes\":{\"opacity\":[0,1]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-83bd2276-96c5-4661-96c6-2952297f0703\",\"keyframes\":{\"transform\":[\"scale(0.3333333333333333)\",\"scale(1)\"]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"forwards\"}]<\/script><\/amp-story-animation><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\" style=\"background-color:#0f1628\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:0;top:-9.25926%;width:100%;height:118.51852%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-ab0e5301-8470-4787-86b2-eed9aac7a633\">\n<div style=\"position:absolute;width:100%;height:100%;left:0%;top:0%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/24-1.jpg\" alt=\"24\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/24-1.jpg 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/24-1-576x1024.jpg 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/24-1-169x300.jpg 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/24-1-150x267.jpg 150w\" sizes=\"(min-width: 1024px) 45vh, 100vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:8.73786%;top:9.70874%;width:82.76699%;height:10.35599%;opacity:1\">\n<div id=\"anim-7f909712-243b-4be9-970f-737b178ce317\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:0;--initial-transform:none\">\n<div id=\"anim-83bd2276-96c5-4661-96c6-2952297f0703\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:scale(0.3333333333333333)\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:0.5865102639296188% 0.5865102639296188% 0.5865102639296188% 0.5865102639296188% \/ 3.125% 3.125% 3.125% 3.125%\" id=\"el-f78ac429-3172-482c-b154-8479e3a74e85\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:-0.5032258064516122% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.388350em;line-height:1.4;text-align:center;padding:0;color:#000000\"><span><span style=\"font-weight: 700; color: #edf6f6\">Mathematical&nbsp;statement&nbsp;of&nbsp;<\/span><br \/>\n<span style=\"font-weight: 700; color: #edf6f6\">Rolle&#8217;s Theorem is<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:9.95146%;top:34.30421%;width:80.09709%;height:38.99676%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-409438d8-b19a-4441-a48e-4ba7a374e6a0\">\n<div style=\"position:absolute;width:100%;height:243.43018%;left:0%;top:-32.08994%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-7-1.png\" alt=\"Rolle&#x27;s theorem-7-1\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-7-1.png 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-7-1-576x1024.png 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-7-1-169x300.png 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-7-1-150x267.png 150w\" sizes=\"(min-width: 1024px) 36vh, 80vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><\/amp-story-page><amp-story-page id=\"1c3ea36c-18bc-49a1-9bbc-9b5da2d5bc6f\" auto-advance-after=\"9s\"><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-34751a32-234b-43ab-a958-215f77c240e6\",\"keyframes\":{\"opacity\":[0,1]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-92efd606-2f37-4793-8a56-f58f42ae6573\",\"keyframes\":{\"transform\":[\"scale(0.3333333333333333)\",\"scale(1)\"]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"forwards\"}]<\/script><\/amp-story-animation><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\" style=\"background-color:#1e1517\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:0;top:-9.25926%;width:100%;height:118.51852%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-0e0c62ee-c080-4e8b-ac38-7d946bf8b34f\">\n<div style=\"position:absolute;width:100%;height:100%;left:0%;top:0%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/40.jpg\" alt=\"40\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/40.jpg 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/40-576x1024.jpg 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/40-169x300.jpg 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/40-150x267.jpg 150w\" sizes=\"(min-width: 1024px) 45vh, 100vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:8.25243%;top:15.37217%;width:76.94175%;height:5.17799%;opacity:1\">\n<div id=\"anim-34751a32-234b-43ab-a958-215f77c240e6\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:0;--initial-transform:none\">\n<div id=\"anim-92efd606-2f37-4793-8a56-f58f42ae6573\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:scale(0.3333333333333333)\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:0.6309148264984227% 0.6309148264984227% 0.6309148264984227% 0.6309148264984227% \/ 6.25% 6.25% 6.25% 6.25%\" id=\"el-089f5cf5-9da6-4879-8e3e-64d85e45c158\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:-0.5638801261829658% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.404531em;line-height:1.4;text-align:center;padding:0;color:#000000\"><span><span style=\"color: #edf6f6\">Example:<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:3.8835%;top:29.28803%;width:92.23301%;height:39.15858%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-d9b5deed-1f6e-4372-9186-0bb8544d2aea\">\n<div style=\"position:absolute;width:109.89129%;height:306.7672%;left:-4.80472%;top:-40.56029%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-Theorem-8-1.png\" alt=\"Rolle&#x27;s Theorem-8-1\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-Theorem-8-1.png 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-Theorem-8-1-576x1024.png 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-Theorem-8-1-169x300.png 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-Theorem-8-1-150x267.png 150w\" sizes=\"(min-width: 1024px) 42vh, 92vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><\/amp-story-page><amp-story-page id=\"91b293a0-73af-4da2-98d8-f7f1ca68d58c\" auto-advance-after=\"9s\"><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-829cdfa8-9721-4724-9633-8f93446426c2\",\"keyframes\":{\"opacity\":[0,1]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-11c64c64-7be9-447d-8248-84e37a8ab091\",\"keyframes\":{\"transform\":[\"scale(0.3333333333333333)\",\"scale(1)\"]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"forwards\"}]<\/script><\/amp-story-animation><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\" style=\"background-color:#fff\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:0;top:-9.25926%;width:100%;height:118.51852%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-c8ddb3ed-4e83-416d-b23c-12f2e7b2c013\">\n<div style=\"position:absolute;width:100%;height:100%;left:0%;top:0%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/3-10.jpg\" alt=\"3 (10)\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/3-10.jpg 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/3-10-576x1024.jpg 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/3-10-169x300.jpg 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/3-10-150x267.jpg 150w\" sizes=\"(min-width: 1024px) 45vh, 100vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:26.69903%;top:18.28479%;width:46.60194%;height:5.17799%;opacity:1\">\n<div id=\"anim-829cdfa8-9721-4724-9633-8f93446426c2\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:0;--initial-transform:none\">\n<div id=\"anim-11c64c64-7be9-447d-8248-84e37a8ab091\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:scale(0.3333333333333333)\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:1.0416666666666665% 1.0416666666666665% 1.0416666666666665% 1.0416666666666665% \/ 6.25% 6.25% 6.25% 6.25%\" id=\"el-6d49e5dc-48a6-430b-ae65-7008ca2bbfa9\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:-0.930989583333334% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.404531em;line-height:1.4;text-align:center;padding:0;color:#000000\"><span><span style=\"color: #edf6f6\">Continuation&#8230;<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:1.45631%;top:22.65372%;width:98.78641%;height:69.7411%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-63f3d2d8-673a-46f3-8d71-157dd822e054\">\n<div style=\"position:absolute;width:100%;height:167.87832%;left:0%;top:-16.01476%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-9-1.png\" alt=\"Rolle&#x27;s theorem-9-1\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-9-1.png 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-9-1-576x1024.png 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-9-1-169x300.png 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Rolles-theorem-9-1-150x267.png 150w\" sizes=\"(min-width: 1024px) 44vh, 99vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><\/amp-story-page><amp-story-page id=\"6c262af1-296f-4440-af05-0482f77ba790\" auto-advance-after=\"9s\"><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-09741cd1-d45b-4d9c-a9b0-d8e1329c27e7\",\"keyframes\":{\"opacity\":[0,1]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"both\"}]<\/script><\/amp-story-animation><amp-story-animation layout=\"nodisplay\" trigger=\"visibility\"><script type=\"application\/json\">[{\"selector\":\"#anim-99fe1256-7ea9-4ff3-8050-342f81d9f1a2\",\"keyframes\":{\"transform\":[\"scale(0.3333333333333333)\",\"scale(1)\"]},\"delay\":0,\"duration\":1300,\"easing\":\"cubic-bezier(.3,0,.55,1)\",\"fill\":\"forwards\"}]<\/script><\/amp-story-animation><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\" style=\"background-color:#041f2a\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:0;top:-9.25926%;width:100%;height:118.51852%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-ab0e5301-8470-4787-86b2-eed9aac7a633\">\n<div style=\"position:absolute;width:100%;height:100%;left:0%;top:0%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/2-7.jpg\" alt=\"2 (7)\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/2-7.jpg 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/2-7-576x1024.jpg 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/2-7-169x300.jpg 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/05\/2-7-150x267.jpg 150w\" sizes=\"(min-width: 1024px) 45vh, 100vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><amp-story-grid-layer template=\"vertical\" aspect-ratio=\"412:618\" class=\"grid-layer\"><\/p>\n<div class=\"page-fullbleed-area\">\n<div class=\"page-safe-area\">\n<div style=\"position:absolute;pointer-events:none;left:11.40777%;top:70.06472%;width:77.42718%;height:24.43366%;opacity:1\">\n<div id=\"anim-09741cd1-d45b-4d9c-a9b0-d8e1329c27e7\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:0;--initial-transform:none\">\n<div id=\"anim-99fe1256-7ea9-4ff3-8050-342f81d9f1a2\" class=\"animation-wrapper\" style=\"width:100%;height:100%;display:block;position:absolute;top:0;left:0;--initial-opacity:1;--initial-transform:scale(0.3333333333333333)\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0;border-radius:0.6269592476489028% 0.6269592476489028% 0.6269592476489028% 0.6269592476489028% \/ 1.3245033112582782% 1.3245033112582782% 1.3245033112582782% 1.3245033112582782%\" id=\"el-d0b8f0e3-4510-4712-9862-3a714bdfa7a5\">\n<p class=\"fill text-wrapper\" style=\"white-space:pre-wrap;overflow-wrap:break-word;word-break:break-word;margin:-0.4931034482758622% 0;font-family:&quot;Merriweather&quot;,serif;font-size:0.355987em;line-height:1.4;text-align:center;padding:0;color:#000000\"><span><span style=\"color: #edf6f6\">Did you find the statement, proof and examples of Rolle\u2019s theorem easy? Help your friends by sharing these important notes.<\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"position:absolute;pointer-events:none;left:-3.8835%;top:-8.57605%;width:107.76699%;height:94.82201%;opacity:1\">\n<div style=\"pointer-events:initial;width:100%;height:100%;display:block;position:absolute;top:0;left:0;z-index:0\" class=\"mask\" id=\"el-a9319910-3fbc-4096-8707-f984994bb6e6\">\n<div style=\"position:absolute;width:100%;height:134.69852%;left:0%;top:-17.34926%\" data-leaf-element=\"true\"><amp-img layout=\"fill\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Jee-Mains-7-1-1.png\" alt=\"Jee-Mains-7-1 (1)\" srcSet=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Jee-Mains-7-1-1.png 720w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Jee-Mains-7-1-1-576x1024.png 576w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Jee-Mains-7-1-1-169x300.png 169w,https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2022\/06\/Jee-Mains-7-1-1-150x267.png 150w\" sizes=\"(min-width: 1024px) 48vh, 108vw\" disable-inline-width=\"true\"><\/amp-img><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><\/amp-story-grid-layer><\/amp-story-page><\/amp-story><\/body><\/html><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In Class 12, \u2018Rolle&#8217;s Theorem\u2019 is a variant of the mean value theorem that satisfies certain conditions.<\/p>\n","protected":false},"author":1,"featured_media":187942,"template":"","meta":{"web_stories_publisher_logo":224645,"web_stories_poster":[],"web_stories_products":[],"footnotes":""},"web_story_category":[2144,2352],"web_story_tag":[3455],"class_list":["post-185033","web-story","type-web-story","status-publish","has-post-thumbnail","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Rolle\u2019s Theorem Class 12 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style=\"color: #edf6f6\">So, let us begin with learning the \u2018Lagrange's Mean Value Theorem\u2019 (LMVT).<\/span>","fontWeight":400,"borderRadius":{"locked":true,"topLeft":2,"topRight":2,"bottomRight":2,"bottomLeft":2},"x":48,"y":56,"width":315,"height":112,"scale":100,"focalX":50,"focalY":50,"marginOffset":-8.725000000000001,"basedOn":"444eed05-ec34-4d0a-95db-c0405443acb3","id":"8c50ba34-8a0e-4688-afa1-32033aca6ab3"},{"opacity":100,"flip":{"vertical":false,"horizontal":false},"rotationAngle":0,"lockAspectRatio":false,"scale":100,"focalX":50,"focalY":50,"resource":{"baseColor":"#fcfc94","blurHash":"U14.9,Rk01%L~pRjIVt600ax_2M{00%M^*M|","type":"image","mimeType":"image\/png","creationDate":"2022-06-06T06:48:06","src":"https:\/\/www.aakash.ac.in\/blog\/wp-content\/uploads\/2022\/06\/Rolles-theorem-1-1.png","width":720,"height":1280,"id":185112,"alt":"Rolle's 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style=\"color: #edf6f6\">The first conditions are the same for both LMVT and Rolle\u2019s Theorem, but the third special condition for Rolle\u2019s theorem is:<\/span>","fontWeight":400,"borderRadius":{"locked":true,"topLeft":2,"topRight":2,"bottomRight":2,"bottomLeft":2},"x":48,"y":468,"width":316,"height":150,"scale":100,"focalX":50,"focalY":50,"marginOffset":-7.677999999999997,"basedOn":"1da4d649-875a-44a4-9e58-4d76dd15e723","id":"92d7383e-9b56-45f4-b90e-305e960e03c5"},{"opacity":100,"flip":{"vertical":false,"horizontal":false},"rotationAngle":0,"lockAspectRatio":false,"scale":100,"focalX":50,"focalY":39.20862962224686,"resource":{"baseColor":"#ffffff","blurHash":"U57BAm_39FIU~qxuM{ay00D%xuxu4nRjxuay","type":"image","mimeType":"image\/png","creationDate":"2022-06-08T10:59:56","src":"https:\/\/www.aakash.ac.in\/blog\/wp-content\/uploads\/2022\/06\/Rolles-theorem-6-1.png","width":720,"height":1280,"id":187937,"alt":"Rolle's 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style=\"font-weight: 700; color: #edf6f6\">Mathematical&nbsp;statement&nbsp;of&nbsp;<\/span>\n<span style=\"font-weight: 700; color: #edf6f6\">Rolle's Theorem is<\/span>","fontWeight":400,"borderRadius":{"locked":true,"topLeft":2,"topRight":2,"bottomRight":2,"bottomLeft":2},"x":36,"y":60,"width":341,"height":64,"scale":100,"focalX":50,"focalY":50,"marginOffset":-7.677999999999997,"basedOn":"06656428-92dd-40c2-9777-728767a8faf3","id":"f78ac429-3172-482c-b154-8479e3a74e85"},{"opacity":100,"flip":{"vertical":false,"horizontal":false},"rotationAngle":0,"lockAspectRatio":false,"scale":100,"focalX":50,"focalY":33.722167912138666,"resource":{"baseColor":"#ffffff","blurHash":"U66RM%%MD%WB~qt7IUj[9FM{xuof00M{%May","type":"image","mimeType":"image\/png","creationDate":"2022-06-08T11:01:20","src":"https:\/\/www.aakash.ac.in\/blog\/wp-content\/uploads\/2022\/06\/Rolles-theorem-7-1.png","width":720,"height":1280,"id":187939,"alt":"Rolle's 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