In this chapter, students can learn how to understand various theorems and where exactly to use them while solving problems. So, two key theorems are being discussed in this chapter. One is Rolle's theorem, and the other is Lagrange's mean value theorem.
Rolle's theorem is one of the most popular mean value theorems out there. It states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f a=f (b), then f'x=0 for some x with a ≤x ≤b. In simple words, if a continuous curve passes through the same y value twice and has a unique tangent line at every point of the interval, then somewhere between the endpoints, it has a tangent parallel to the x-axis.
After understanding Rolle's theorem, students get to study the geometrical interpretation of Rolle's theorem, Algebraic interpretation of Rolle's theorem and learn how to check the applicability of Rolle's theorem. Verifications of Rolle's theorem for a given function defined within a given interval. And also the miscellaneous applications of Rolle's theorem.
Furthermore, students get their eyes on Lagrange's mean value theorem. It states that if a function f is described on the closed interval [a, b] satisfying the following two conditions, one is the function f(x) is continuous on the closed interval [a, b], the other one is the function if is differentiable on the open interval (a, b). Then, there exists a value noted as x = c in the following way,
f'c=f b-f ab-a
So, this theorem can also be termed as the first mean value theorem.
Students get to study the geometrical interpretation and verification, proving inequalities and miscellaneous applications of Lagrange's mean value theorem like they did with Rolle's theorem.