Extrema using Second Derivative Test

The Second Derivative Test relates critical points, extreme values, and concavity into a single practical tool for classifying whether a critical point is a relative minimum or maximum.

Theorem 7.13 (The Second Derivative Test). Suppose $c$ is a critical point at which $f'(c)=0$, $f'(x)$ exists in a neighborhood of $c$, and $f''(c)$ exists. Then $f$ has:

• a relative maximum value at $c$ if $f''(c)<0$;
• a relative minimum value at $c$ if $f''(c)>0$;
• no information from this test if $f''(c)=0$ — the test is inconclusive, and the classification must fall back on the First Derivative Test (§7.6.4) instead.

Worked pattern (as in Examples 7.59–7.61). Find the critical numbers by solving $f'(x)=0$; compute $f''(x)$ and evaluate it at each critical number; apply the sign rule above to classify each one, falling back to a sign table of $f'$ wherever $f''=0$. …