Second Derivative Test

Classify a critical point as a local max or min by checking the sign of the second derivative there instead of watching for a sign change.

At a critical point $c$ where $f'(c)=0$ : if $f''(c)>0$ , $f$ has a local minimum at $c$ ; if $f''(c)<0$ , a local maximum. If $f''(c)=0$ , the test is inconclusive and the First Derivative Test must be used instead.

Examples

For $f(x)=x^3-3x$ : $f''''(x)=6x$ , so $f''''(1)=6>0$ (local min at $x=1$ ) and $f''''(-1)=-6<0$ (local max at $x=-1$ ).