Stationary Points & Curve Sketching

This lesson covers finding stationary points, classifying them, identifying points of inflection, and developing a systematic approach to curve sketching. These are essential skills for A-Level Mathematics and are examined in both pure and applied contexts.

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Stationary Points

A stationary point occurs where the gradient of a curve is zero:

dy/dx = 0

At a stationary point, the tangent to the curve is horizontal.

Types of Stationary Point

1. Local maximum — the curve rises to the point then falls.
2. Local minimum — the curve falls to the point then rises.
3. Stationary point of inflection — the curve flattens momentarily but continues in the same direction (either rising–flat–rising or falling–flat–falling).

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The First Derivative Test

Examine the sign of dy/dx on either side of the stationary point:

Sign change of dy/dx	Type
+ → 0 → −	Local maximum
− → 0 → +	Local minimum
+ → 0 → +	Rising point of inflection
− → 0 → −	Falling point of inflection

Example 1