Second derivative test

The second derivative test is a shortcut for deciding whether a critical point is a local maximum or a local minimum. Start by finding the critical points — values of c where f′(c) = 0 — then evaluate the second derivative at each one. A positive f″(c) means the graph is concave up at c, so the curve turns upward and c is a local minimum. A negative f″(c) means the graph is concave down, so c is a local maximum.

Take f(x) = x³ − 3x. Then f′(x) = 3x² − 3, which is zero at x = −1 and x = 1. Since f″(x) = 6x, we get f″(−1) = −6 < 0, a local maximum, and f″(1) = 6 > 0, a local minimum. The whole classification takes one substitution per critical point.

The test has one real limitation: when f″(c) = 0 it gives no information, and you must fall back on the first derivative test, which examines whether f′ changes sign around c. Both f(x) = x⁴ and f(x) = x³ have f′(0) = 0 and f″(0) = 0, yet x = 0 is a minimum for the first and an inflection point for the second. The second derivative is also what identifies concavity and inflection points — places where f″ changes sign and the curve switches from concave up to concave down.

The second derivative test is required content on the AP Calculus AB exam, where it appears in the analytical applications unit alongside concavity and the first derivative test. It also shows up in the mathematics sections of the FE Mechanical and FE Civil exams, where differential calculus is used to locate maxima and minima in engineering optimization problems.