## First Derivative Test

Definition. Let $f\colon D\to\mathbb{R}$ be a function. Then a “Local Minima” is a point $c\in D$ such that $f(c)\leq f(x)$ for any $x$ in some open interval containing $c$.

Likewise a “local maxima” is a point $c\in D$ which has $f(c)\geq f(x)$ for every $x$ in some open interval containing $c$.

First Derivative Test. If $f$ has a local maxima (or minima) at an interior point $c$ of its domain, and if $f'(x)$ is defined on $c$, then $f'(c)=0$.

Proof. We will consider the case for a local minima (the reasoning is very similar for a local maxima). If $x>c$, then $x-c>0$. We see that
(1)$\displaystyle\frac{f(x)-f(c)}{x-c}\geq0$
On the other hand, if $x, then $x-c<0$. We see that
(2)$\displaystyle\frac{f(x)-f(c)}{x-c}\leq0$
Thus by the squeeze theorem we see that
(3)$\displaystyle 0\leq\lim_{x\to c}\frac{f(x)-f(c)}{x-c}\leq0$
implies $f'(c)=0$.

Remark. Observe the contrapositive of this theorem states: if $f'(c)\not=0$, then $c$ cannot be a local maxima or local minima for $f$.