Second Derivative Test

The central idea

Suppose $f'(c)=0$ and $f''$ exists near $c$. If $f''(c)>0$, then $f$ has a local minimum at $c$; if $f''(c)<0$, it has a local maximum. If $f''(c)=0$ or does not exist, the test is inconclusive, not proof of no extremum.

Definitions, hypotheses, and notation

The test is a sufficient classification rule, not a universal one. At $f(x)=x^4$, both $f'(0)$ and $f''(0)$ vanish, yet the point is a strict local minimum. Higher-order terms or a first-derivative sign chart reveal what the zero second derivative cannot. Similarly, points where $f'$ is undefined must be classified by another method.

For a twice-differentiable function, the sign of $f''(c)$ describes how slopes change through a stationary point. A negative value means slopes are decreasing and cross from positive to negative nearby; a positive value means the reverse. The theorem packages that local sign behavior, while closed-interval absolute extrema still require endpoint comparison.

Conceptual meaning

At a stationary point, positive second derivative means the graph bends upward like a cup, placing the point locally low. Negative second derivative means it bends downward like a cap. The first-derivative condition is indispensable.

A dependable method and decision rule

1. Find critical numbers from $f'=0$ or undefined within the domain.
2. Use this test only at critical numbers where $f'(c)=0$ and $f''(c)$ can be evaluated.
3. Compute $f''(c)$ and classify by its sign.
4. When the result is zero or undefined, switch to the First Derivative Test.
5. Report the point $(c,f(c))$ and whether the extremum is local.

Fully worked example

Common mistakes and why they fail