Unlocking The Power Of Rolle’S Theorem: Exploring The Fundamental Calculus Theorem For Finding Roots And Extreme Values

Rolle’s Theorem

if f(x) is continuous on [a,b] and differentiable on (a,b), and f(a)=f(b), then there is a number c between (a,b) where f'(c) = 0

Rolle’s Theorem is a fundamental theorem in calculus named after Michel Rolle, a French mathematician. It is a theorem that deals with differentiable functions, and its statement is as follows:

Suppose that f(x) is a continuous function on the closed interval [a, b], and is differentiable on the open interval (a, b). If f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.

In simpler terms, Rolle’s Theorem states that if a continuous function f(x) takes on the same value at two different points a and b, then there must be at least one point c between a and b where the function’s derivative f'(c) is equal to zero.

The theorem is particularly useful in applications where one wants to find the root of a function, that is, find a point where the function equals zero. Knowing that the function’s derivative is zero at least once, allows us to locate these roots.

One application of Rolle’s Theorem is finding the extreme values of a function. If a differentiable function f(x) has a maximum or minimum value at an interior point c, then f'(c) must be zero. Rolle’s Theorem provides a way to guarantee that such a point exists, and we can use this information to locate the extreme points of a function.

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