5.7 The Second Derivative Test

The test

At a critical point where f'(c) = 0: if f''(c) > 0, c is a local minimum (concave up); if f''(c) < 0, c is a local maximum (concave down).

Inconclusive case

If f''(c) = 0, the test is inconclusive — fall back on the First Derivative Test.

Key terms

Second Derivative Test

Classify a critical point using f''(c).

f''(c) > 0

Local minimum (concave up).

f''(c) < 0

Local maximum (concave down).

f''(c) = 0

Inconclusive — use the First Derivative Test.

Basis

Concavity at the critical point.

Try an AP-style question

For f(x) = x³ − 3x, what does the second derivative test say at x = 1?

1. Inconclusive, since f(1) < 0
2. Inflection point, since f'(1) = 0
3. Relative minimum, since f''(1) = 6 > 0
4. Relative maximum, since f''(1) = 6 > 0