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MathematicsA Level & IB

What is stationary point?

A stationary point is a point on a curve where the gradient is zero — a maximum, a minimum or a point of inflection.

Stationary point explained

Find stationary points by solving dydx=0\frac{dy}{dx} = 0. To classify them, use the second derivative: negative means a maximum, positive means a minimum, and zero means you must test the gradient either side.
Stationary points underpin optimisation questions, where you model a quantity, differentiate, set the derivative to zero and justify that your answer really is the maximum or minimum.

Key formula

dydx=0\dfrac{dy}{dx} = 0, then test d2ydx2\dfrac{d^{2}y}{dx^{2}}

Worked example

For y=x3−3xy = x^{3} - 3x, dydx=3x2−3=0\frac{dy}{dx} = 3x^{2} - 3 = 0 gives x=±1x = \pm 1. Since d2ydx2=6x\frac{d^{2}y}{dx^{2}} = 6x, x=1x = 1 is a minimum and x=−1x = -1 a maximum.

Examiner tip

Give the full coordinates, not just the x-value, and state the nature of each point explicitly — both are separate marks.

Stationary point: common questions

What if the second derivative is zero?
The test is inconclusive. Check the sign of the first derivative just before and just after the point to decide between maximum, minimum and inflection.
Is every point of inflection a stationary point?
No. A point of inflection is where the curve changes concavity; only some of them also have zero gradient.

Related terms

Differentiation

Differentiation is the process of finding the rate at which one quantity changes with respect to another — the gradient of a curve at a point.

Gradient

The gradient of a line measures its steepness — the change in y divided by the change in x between two points.

Second derivative

The second derivative is the derivative of the derivative — it measures how fast the gradient itself is changing, and hence the concavity of a curve.

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