5.7 Using the Second Derivative Test to Determine Extrema

Second Derivative Test

If f'(c) = 0 and f''(c) exists, check the sign of f''(c): if f''(c) > 0, the graph is concave up and c is a local minimum; if f''(c) < 0, the graph is concave down and c is a local maximum. This is often faster than the first derivative test when f'' is easy to compute.

When the Test Fails

If f''(c) = 0 or f''(c) is undefined, the second derivative test gives no conclusion — you must fall back on the first derivative test (sign change of f' around c) to classify the critical point. This inconclusive case often occurs at inflection points or higher-order critical points like x⁴.

Connecting Concavity and Extrema

The second derivative test links concavity directly to local extrema: local min occurs where the curve is concave up around a critical point, local max where it's concave down. Always confirm f'(c) = 0 first — the test only classifies critical points, it doesn't find them.

Key terms

Second Derivative Test (min case)

If f'(c) = 0 and f''(c) > 0, then f has a local minimum at c (concave up).

Second Derivative Test (max case)

If f'(c) = 0 and f''(c) < 0, then f has a local maximum at c (concave down).

When test is inconclusive

If f''(c) = 0 or is undefined, use the first derivative test instead.

Prerequisite for using the test

c must already be a critical point (f'(c) = 0 or undefined); the test only classifies it.

Concavity and extrema link

Concave up at a critical point → local min; concave down at a critical point → local max.

Try an AP-style question

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

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