Local maxima and minima

A local maximum is a hilltop: higher than the points just around it. A local minimum is a valley bottom. 'Local' means only nearby: another hill elsewhere may be higher.

At a smooth hilltop or valley, the tangent is flat, so f′(c) = 0. Points where f′(c) = 0 (or f′ does not exist) are called critical points. They are the candidates.

First derivative test: look at how the sign of f′ changes as you pass through c.

• + to – (up, then down): local maximum.
• – to + (down, then up): local minimum.
• no change: neither. It is a point of inflexion, like x³ at 0.

Second derivative test (often quicker): at a point where f′(c) = 0,

• f″(c) < 0: the curve bends down like a cap, so maximum.
• f″(c) > 0: the curve bends up like a cup, so minimum.
• f″(c) = 0: the test fails; go back to the first derivative test.

Example: f(x) = x³ – 6x² + 9x + 1. f′(x) = 3(x – 1)(x – 3), zero at 1 and 3. f″(x) = 6x – 12. f″(1) = –6 < 0, so a local max f(1) = 5. f″(3) = 6 > 0, so a local min f(3) = 1.

When the test fails: f(x) = x⁴ has f″(0) = 0, but f′ goes from – to + at 0, so 0 is a local minimum.

A function with no critical points, such as eˣ or log x, has no local maxima or minima at all: it only goes one way.