Increasing, Decreasing and Extrema

Reading f from f ′

$f'>0$ on an interval $\Rightarrow$ $f$ is increasing there; $f'<0$ $\Rightarrow$ decreasing. Critical points are where $f'=0$ or $f'$ is undefined.

First Derivative Test. At a critical point $c$: if $f'$ changes $+\to-$, $f$ has a local maximum; if $-\to+$, a local minimum; if no sign change, neither.

Candidates test for absolute extrema on a closed interval $[a,b]$: evaluate $f$ at every critical point and at both endpoints, then compare the values. Endpoints are the step most often forgotten.

Justification language matters. "$f'(2)=0$ so there is a maximum" earns nothing; "$f'$ changes from positive to negative at $x=2$, so $f$ has a local maximum there" earns the point.