Proofs of Derivative Applications Facts — Question 8

Question 8 - Solution

By the definition of the second derivative and $f'(a)=0$,

$$\lim_{h\to0}\frac{f'(a+h)}h=f''(a).$$

Because this limit is nonzero, the quotient has the same sign as $f''(a)$ for all sufficiently small nonzero $h$.

Consequently, $f'(a+h)<0$ for negative $h$ and $f'(a+h)>0$ for positive $h$. Thus $f$ is strictly decreasing just left of $a$ and strictly increasing just right of $a$, by the Mean Value Theorem.

Therefore $f$ has a strict local minimum at $a$. No continuity assumption on $f''$ near $a$ is needed.