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 
(don’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.
