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.
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.