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Rolle's Theorem and the Mean Value Theorem

Rolle's Theorem

Rolle's theorem states:

Given  f is continuous on closed interval [a,b], if f(a) = f(b), then there is at least one number c s.t. f'(c) = 0.

So, if the two outer points in an interval are the same, there is at least one point where the function has a derivative of zero. Above, I have an image of a semi-circle $y = \sqrt{25 - x^2}$, so $f(-5) = f(5)$, so there must be a point where $f'(x) = 0$, and sure enough, $f'(0) = 0$.

Mean Value Theorem

The Mean Value Theorem, usually abbreviated as MVT, states that if f is continuous on a closed interval [a,b],and is differentiable on the open interval (a,b) then there exists c in (a,b) s.t. $f'(c)$ = average slope, or $\frac{f(b) - f(a)}{b - a}$.
David Witten

(Not) The First Derivative Test

Related Rates Sphere Problem