First Derivative Test

Classify a critical point as a local max, local min, or neither by watching whether $f'$ changes sign there.

At a critical point $c$ , the First Derivative Test says: if $f'$ changes from positive to negative at $c$ , $f$ has a local maximum there; from negative to positive, a local minimum; if $f'$ keeps the same sign on both sides, $c$ is neither.

Examples

For $f(x)=x^3-3x$ : $f''$ is positive then negative at $x=-1$ (local max), negative then positive at $x=1$ (local min).