Higher Derivatives and Concavity

The Second Derivative

Differentiating a derivative gives the second derivative $f''(x)$, the rate of change of the rate of change. If $f$ is position, $f'$ is velocity and $f''$ is acceleration. Higher derivatives ($f'''$, etc.) continue this pattern.

The second derivative reveals concavity, how a curve bends:

A point where concavity changes sign is an inflection point. While $f''(x) = 0$ is necessary, it is not sufficient: concavity must actually switch signs.

Tests and Curve Sketching

Concavity gives the second-derivative test for critical points where $f'(x) = 0$:

Pitfall: $f''(x) = 0$ does not guarantee an inflection point. For $f(x) = x^4$, $f''(0) = 0$, but it is concave up everywhere, so $x=0$ is not an inflection point.

Together, both derivatives determine shape, driving curve sketching and optimization.