Mean Value Theorem

Mean Value Theorem: Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists $c\in(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}.$

This theorem links global change (the secant slope) to local change (a derivative ). It is the main engine behind monotonicity results, error estimates, and many uniqueness arguments.

Proof sketch: Define $g(x)=f(x)-\frac{f(b)-f(a)}{b-a}(x-a).$ Then $g$ is continuous on $[a,b]$, differentiable on $(a,b)$, and satisfies $g(a)=g(b)$. By Rolle's theorem , there is $c\in(a,b)$ with $g'(c)=0$, which rearranges to the desired formula.