Discovering Critical Points With Rolle’S Theorem: A Useful Tool In Calculus

Rolle’s Theorem

f is continuous on [a,b]differentiable on (a.b)f(a)=f(b)there is a number c between them such that f'(c)=0*specific case of MVT where endpoints are equal

Rolle’s Theorem is a theorem in calculus that is used to determine whether a function has a critical point or not. It states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on (a,b), and if f(a) = f(b), then there is at least one point c in (a,b) such that f'(c) = 0.

This theorem is useful in finding the roots of a function when there are several of them, as it tells us that there must be at least one point where the derivative of the function is zero. In practice, we can use this theorem to find the critical points of a function, which are the points where the derivative is zero, or where it does not exist.

It is important to note that this theorem only tells us that a critical point exists, but it does not tell us anything about the nature of the point. For instance, it does not tell us whether it is a maximum, minimum, or inflection point. However, it does provide us with a useful starting point for further investigation and analysis of the function.

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