Mean Value Theorem
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Mathematics/Calculus
Rolle's TheoremTheorem 1. Let $f$ be a function that satisfies the following conditions:(1) $f$ is continuous on $[a, b]$.(2) $f$ is differentiable on $(a, b)$(3) $f(a) = f(b)$Then there is a number $c \in (a, b)$ such that $f'(c) = 0$.Proof. We may think of three cases. (1) If $f(x) = k$ for any constant $k$, then $c$ can be taken to be any number in $(a, b)$.(2) If $f(x) > f(a), \forall x \in ..