Year 12 Calculus: Stationary points

Stationary points occur where the derivative is zero. They are important because they help you identify local maxima, local minima and turning behavior in graphs and optimization problems.

Do not stop when you solve

f' (x) = 0

. You still need to classify the point or interpret it in context.

Worked example

Problem: Find the stationary points of

f (x) = x^{3} - 3 x^{2} + 2

Differentiate: $f' (x) = 3 x^{2} - 6 x$ .
Set the derivative equal to zero: $3 x^{2} - 6 x = 0$ .
Factor: $3 x (x - 2) = 0$ .
So the stationary x-values are $x = 0$ and $x = 2$ .
Find the y-values: $f (0) = 2$ and $f (2) = 8 - 12 + 2 = - 2$ .

Answer: the stationary points are

(0, 2)

and

(2, - 2)

What a stationary point means

A stationary point is a point where the tangent is horizontal, so the derivative is zero. That does not automatically tell you whether the point is a maximum, a minimum, or a stationary point of inflection. To finish the job, you need either a second-derivative check or a sign analysis of $f' (x)$ around the point.

Worked example 2

Problem: Classify the stationary points of

g (x) = x^{3} - 3 x

Differentiate: $g' (x) = 3 x^{2} - 3 = 3 (x - 1) (x + 1)$ .
Set $g' (x) = 0$ , giving $x = - 1$ and $x = 1$ .
Find the coordinates: $g (- 1) = 2$ and $g (1) = - 2$ .
Use the second derivative: $g'' (x) = 6 x$ .
At $x = - 1$ , $g'' (- 1) = - 6$ , so $(- 1, 2)$ is a local maximum.
At $x = 1$ , $g'' (1) = 6$ , so $(1, - 2)$ is a local minimum.

Answer:

(- 1, 2)

is a local maximum and

(1, - 2)

is a local minimum.

Common mistakes

Listing only the x-values and forgetting the coordinates.
Calling every stationary point a turning point without checking.
Ignoring the wording of the question after finding the points.

Stationary-point questions are one of the main ways calculus connects to graph sketching and optimisation. A derivative equal to zero signals a possible turning behaviour, but the mathematical job is not finished until you decide what kind of behaviour it is. That is why classification matters so much. In modelling questions, the classification is often the part that gives the answer meaning.

A good revision habit is to practise describing the result in words after the algebra is complete. For example, do not stop at "x equals 1". Say "there is a local maximum at x equals 1" or "this stationary point gives the smallest area". That final sentence is what ties the derivative work back to the problem.

Revision focus

In calculus, one of the best habits is to decide what role the expression is playing before you touch a rule. Is it a product, a composite function, an implicitly defined curve, or an antiderivative problem? That classification step is not wasted time. It is the step that usually determines whether the next line of working will be short and accurate or long and unstable. Strong calculus work looks calm because the structure is identified early, not because the algebra is magically easier.