Rolle’S Theorem Explained: The Conditions For Critical Points In Calculus

Rolle’s Theorem

if f(x) is continuous on [a,b] and differentiable on (a,b), and if f(a)=f(b), then there is at least one point (x=c) on (a,b) where f'(c)=0

Rolle’s Theorem is a basic result in calculus that deals with the conditions required for a differentiable function to have a critical point (i.e., maximum or minimum). The theorem is named in honor of the French mathematician Michel Rolle.

Formally, Rolle’s Theorem states that if a function f(x) is continuous and differentiable over a closed interval [a, b], and if the values of f(a) and f(b) are equal (i.e., f(a) = f(b)), then there exists at least one point c in the open interval (a, b) such that f'(c) = 0.

In other words, if a function has the same value at its endpoints, then there must be a point between the endpoints where the function is level (i.e., where the derivative is zero).

Rolle’s Theorem can be used to prove important results in calculus, such as the Mean Value Theorem and the First Derivative Test. It is also useful in optimization problems and can help identify solutions to equations with unknown values.

One important restriction to keep in mind when applying Rolle’s Theorem is that it only applies to continuous functions that are differentiable over their entire domain. If a function is discontinuous or has a point where the derivative is undefined, then Rolle’s Theorem may not be applicable.

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