A stationary point on a curve is where the gradient is $0$. This occurs when $f'(x) = 0$. For example, the curve $y = x^2$ has a stationary point at $(0, 0)$.

Local Maximum and Minimum

These are stationary points that represent the highest or lowest points in their immediate area. To find which one it is, we check the gradient slightly before ($x - h$) and slightly after ($x + h$) the point.

Spelling

• The plural of maximum is maxima.
• The plural of minimum is minima.

Point of Inflection

A point of inflection is a stationary point where the gradient is $0$, but the sign of the gradient stays the same on both sides.

Comparison of Stationary Points

This comparison is done by sampling two values left/right of the stationary point, $h$ represents the offset.

The gradient of the point of inflection has the same sign on each side.

Summary

If $\frac{d^{2}y}{dx^{2}} < 0$, then $x$ is the local maximum. If $\frac{d^{2}y}{dx^{2}} > 0$, then $x$ is the local minimum. If $\frac{d^{2}y}{dx^{2}} = 0$ then sample around and compare - this does not mean that it is a point of inflection.