Second Derivative

f'' sets concavity and tests maxima vs minima.

The idea

The second derivative $f''$ is the derivative of the derivative: $f'$ measures the rate at which $f$ changes, and $f''$ measures the rate at which that slope changes. If $f$ is position and $f'$ is velocity, then $f''$ is acceleration — not how fast you are moving, but how fast that speed is changing.

Geometrically, $f''$ measures concavity, the direction in which a curve bends. Where $f'' > 0$ the slope is increasing, so the curve bends upward like a cup; it is concave up. Where $f'' < 0$ the slope is decreasing and the curve bends downward; it is concave down. A point where $f''$ changes sign is a point of inflection, where the bend reverses.

Concavity classifies critical points. At a point $c$ with $f'(c) = 0$ the tangent is horizontal, and the sign of $f''(c)$ decides what kind of point $c$ is: if $f''(c) > 0$ the curve is concave up there, so $c$ is a local minimum, and if $f''(c) < 0$ it is a local maximum. This is the second-derivative test. When $f''(c) = 0$ the test decides nothing, and the sign of $f'$ on each side of $c$ must be checked instead.

Ways to work on it

• Walkthrough. Concavity, inflection points, and the second-derivative test.
• Practice. Evaluate f'' and read off concavity.
• Hardest. Classify both critical points of a cubic.

Not sure where to start? Take the ten-question placement test.