Discover the Power of Rolle’s Theorem: Exploring the Proof and Applications.

Rolle’s Theorem

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.

Rolle’s Theorem is a mathematical theorem that deals with the existence of a point in a function where its derivative is zero. It is named after Michel Rolle, a French mathematician who first proved the theorem in 1691.

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

In simple terms, Rolle’s Theorem states that if a function has the same value at the endpoints of an interval and it is continuous and differentiable over the whole interval, then there must be at least one point in the interval where the derivative of the function is equal to zero.

To illustrate this theorem, consider a function f(x)=x^2-2x+1, which is continuous and differentiable on the interval [0,2]. We can verify that f(0) = f(2) = 1. To apply Rolle’s Theorem, we find the derivative of the function: f'(x) = 2x-2. We can see that f'(x) is equal to zero when x=1. Therefore, there exists at least one point c in the open interval (0, 2) such that f'(c) = 0.

Rolle’s Theorem is often used as a preliminary step in proving other theorems in calculus. It is also helpful in finding roots of polynomial equations and in optimization problems.

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