What are Stationary Points?

Stationary points (or turning/critical points) are the points on a curve where the gradient is 0. This means that at these points the curve is flat. Usually, the gradient of a curve is always changing and so the gradient is only 0 instantaneously (unless the curve is a flat line, in which case, the gradient is always 0).

A MAXIMUM is located at the top of a peak on a curve. Conversely, a MINIMUM if it is at the bottom of a trough.

A stationary point can be found by solving , i.e. finding the x coordinate where the gradient is 0. See more on differentiating to find out how to find a derivative.

See Example 1.

Click here for an online tool for checking your stationary points. Page 21 onwards of the StudyWell Differentiation eGuide has more on Stationary Points including exam-style questions.

Classifying Stationary Points

For certain functions, it is possible to differentiate twice (or even more) and find the second derivative. It is often denoted as or . For example, given that then the derivative is and the second derivative is given by .

The second derivative can tell us something about the nature of a stationary point:

• For a MINIMUM, the gradient changes from negative to 0 to positive, i.e. the gradient is increasing. Hence, the second derivative is positive – .
• For a MAXIMUM, the gradient changes from positive to 0 to negative, i.e. the gradient is decreasing. Hence, the second derivative is negative – .

We can classify whether a point is a minimum or maximum by determining whether the second derivative is positive or negative. This is done by putting the -coordinates of the stationary points into .

Page 21 onwards of the StudyWell Differentiation eGuide has more on Stationary Points including exam-style questions.

Examples

Find the stationary points on the curve .

Solution:

Start by solving :

i.e. . Factorising gives and so the coordinates are and . Substituting these into the equation gives the coordinates of the turning points as and .

Find and classify the stationary points of .

Solution:

We first locate them by solving . is given by

.

We can solve by factorising:

which gives or . The corresponding coordinates are (donâ€™t be afraid of strange fractions) and . Hence, the critical points are at and . We can classify them by substituting the coordinate into the second derivative and seeing if it is positive or negative. Differentiating a second time gives
. It follows that which is less than 0, and hence is a MAXIMUM. Similarly, and is a MINIMUM.

Videos

Using differentiation to locate and classify the minimum of the cost of a journey.

Using stationary points to sketch a functions that is a combination of a reciprocal and a cubic function.

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