The Second Derivative Test

The first derivative test works in every case, but it requires checking the sign of $f'$ on both sides of each critical point. When the function is easy to differentiate twice, a quicker test uses the second derivative $f''(x)$, which measures the rate of change of the slope itself and therefore describes the concavity (bending) of the curve.

Concavity. Where $f''(x) > 0$ the slope is increasing, so the curve bends upward like a cup — it is concave up. Where $f''(x) < 0$ the slope is decreasing and the curve bends downward like a cap — it is concave down. A local minimum sits at the bottom of a concave-up piece; a local maximum sits at the top of a concave-down piece.

The Second Derivative Test. Let $c$ be a critical point, so $f'(c) = 0$. Then:

• if $f''(c) < 0$, the curve is concave down at $c$, so $f$ has a local maximum at $c$;
• if $f''(c) > 0$, the curve is concave up at $c$, so $f$ has a local minimum at $c$;
• if $f''(c) = 0$, the test is inconclusive — fall back on the first derivative test.

Method.

1. Find $f'(x)$ and solve $f'(x) = 0$ for the critical points.
2. Find $f''(x)$ and evaluate it at each critical point.
3. A negative value signals a maximum, a positive value a minimum; a zero value means the test fails and the first derivative test must be used instead.
4. Substitute the critical value into $f$ to get the extreme value. …