Example for Homework 10

elishapeterson 16 Apr 2008 17:55

**Claim:** if $f$ is differentiable on $(a,b)$, continuous on $[a,b]$, and obtains a minimum (or maximum) value at $c\in(a,b)$, then $f'(c)=0$.

**Proof:** Let *c* be a maximum value. Differentiability implies that $\lim_{t\to\theta}\frac{f(t)-f(\theta)}{t-\theta}$ exists. If $t>c$ then $\frac{f(t)-f(c)}{t-c}\leq 0$ since $f(t)\leq f(c)$. If $t<c$ then $\frac{f(t)-f(c)}{t-c}\geq 0$. Therefore, $f'(c)$ can be written as the limit of (i) a sequence of nonnegative numbers, and (ii) a sequence of nonpositive numbers. The only possible limit is therefore 0. $\blacksquare$