What is a stationary point?

A point where the gradient of the curve is zero, so the tangent is horizontal. It can be a maximum, a minimum or a point of inflection.

What if the second derivative is zero?

The test gives no answer. Check the sign of dy/dx just before and just after the point: negative then positive is a minimum, positive then negative is a maximum.

How to Find Stationary Points and Their Nature

Short answer

How do you find stationary points and their nature?

Differentiate and solve dy/dx = 0 to find the x-coordinates, then substitute back for the y-coordinates. To decide the nature, find d²y/dx²: negative means a maximum, positive means a minimum, and zero means you must test the gradient either side.

A stationary point is where the curve is momentarily flat: its gradient $\frac{d y}{d x}$ is zero.

The method

Differentiate to get $\frac{d y}{d x}$ .
Solve $\frac{d y}{d x} = 0$ for the $x$ -coordinates.
Substitute into $y$ for the $y$ -coordinates.
Find $\frac{d ^{2} y}{d x ^{2}}$ at each point: negative is a maximum, positive is a minimum.

Worked example

Find the stationary points of $y = x^{3} - 6 x^{2} + 9 x + 1$ and their nature.

Steps 1 and 2:

\frac{d y}{d x} = 3 x^{2} - 12 x + 9 = 3 (x - 1) (x - 3) = 0 ⟹ x = 1 or x = 3

Step 3: $y (1) = 1 - 6 + 9 + 1 = 5$ and $y (3) = 27 - 54 + 27 + 1 = 1$ .

Step 4:

\frac{d ^{2} y}{d x ^{2}} = 6 x - 12

At $x = 1$ : $6 - 12 = -6 < 0 $, so$ (1, 5)$ is a maximum.
At $x = 3$ : $18 - 12 = 6 > 0 $, so$ (3, 1)$ is a minimum.

When the second derivative is zero

For $y = x^{4}$ , both derivatives are zero at $x = 0$ , so the second-derivative test says nothing. Check the gradient either side instead: $\frac{d y}{d x} = 4 x^{3}$ is negative just before 0 and positive just after, so $(0, 0)$ is a minimum.

Try it: classify the two stationary points

Predict which is the maximum before the lab shows you.

loading lab…

Mistakes that cost marks

Giving only the $x$ -coordinates. A point needs both coordinates.
Reading the second derivative backwards. Negative means maximum (the curve bends down).
Concluding "inflection" from $\frac{d ^{2} y}{d x ^{2}} = 0$ . It only means the test failed; check the gradient either side.

How to find stationary points and their nature