Finding stationary points

Finding stationary points.

You can find stationary points on a curve by differentiating the equation of the curve and finding the points at which the gradient function is equal to 0.

One way of determining a stationary point.

The nature of the stationary point can be found by considering the sign of the gradient on either side of the point.

A stationary point can be:

- A local maximum, where the gradient changes from positive to negative (+ to -)
- A local minimum, where the gradient changes from negative to positive (- to +)
- A stationary point of inflection, where the gradient has the same sign on both sides of the stationary point

An alternative method for determining the nature of stationary points. 

If you differentiate the gradient function, the result is called a second derivative.

At a stationary point:

- If the second derivative is positive, the point is a local maximum
- If the second derivative is negative, the point is a local minimum
- If the second derivative is 0, the stationary point could be a local minimum, a local maximum or a stationary point of inflection.
 – (you need to look at the gradient on either side to find the nature of the stationary point)

Example using the second method:
y = x3 - x2 - 4x -1
find the values of the first and second derivatives where x= -1

dy/dx = 3x2 - 2x - 4 = (3 x -1 x -1) - (2 x -1) - 4 = 1
d2y/dx2 = 6x - 2 = (6 x -1) - 2 = -8

Since the second derivative (d2y/dx2) < 0, the point where x= -1 is a local minimum. 

Jessica F. A Level Economics tutor, A Level Maths tutor, GCSE Maths t...

2 years ago

Answered by Jessica, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Walter T.

Degree: Civil Engineering (Masters) - Bristol University

Subjects offered:Maths, Further Mathematics

Further Mathematics

“Hi! I'm Walter, a specialist maths tutor and a first year civil engineering student at the University of Bristol.”

£20 /hr

Naomi S.

Degree: Natural Sciences (Maths and Physics) (Masters) - Durham University

Subjects offered:Maths, Spanish+ 4 more

Further Mathematics
-Personal Statements-

“Friendly, enthusiastic and committed Maths and Physics student at Durham University. ”

MyTutor guarantee

£26 /hr

Alex B.

Degree: Physics (Masters) - Durham University

Subjects offered:Maths, Physics


“About me: I’m Alex and I have just completed my first year studying physics at Durham University, with a first. I have a passion for mathematics and how it can be applied to solve problems in physics. Throughout my A levels, other stu...”

About the author

£20 /hr

Jessica F.

Degree: Economics (Bachelors) - Exeter University

Subjects offered:Maths, Economics


“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Posts by Jessica

Explain why monopolies may be an undesirable form of market structure

Finding stationary points

When do I use a cosine rule over a sine rule?

Why have inequalities increased in recent years?

Other A Level Maths questions

If cos(x)= 1/3 and x is acute, then find tan(x).

Find the integral of log|x| by integration by parts

How would you differentiate f(x)=3x(2x-1)^2

how to integrate by parts

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss