How do I find the maxima and minima of a function?

What are the maxima and minima?

The maxima of a function f(x) are all the points on the graph of the function which are 'local maximums'. A point where x=a is a local maximum if, when we move a small amount to the left (points with xa), the value of f(x) decreases. We can visualise this as our graph having the peak of a 'hill' at x=a.

Similarly, the minima of f(x) are the points for which, when we move a small amount to the left or right, the value of f(x) increases. We call these points 'local minimums', and we can visualise them as the bottom of a 'trough' in our graph.

One similarity between the maxima and minima of our function is that the gradient of our graph is always equal to 0 at all of these points; at the very top of the peaks and the very bottom of the troughs, the slope of our graph is completely flat. This means our derivative, f '(x), is equal to zero at these points.

How do we find them?

1) Given f(x), we differentiate once to find f '(x).

2) Set f '(x)=0 and solve for x. Using our above observation, the x values we find are the 'x-coordinates' of our maxima and minima.

3) Substitute these x-values back into f(x). This gives the corresponding 'y-coordinates' of our maxima and minima.

Which of these points are maxima and which are minima?

Here we may apply a simple test. Assume we've found a stationary point (a,b):

1) Differentiate f '(x) once more to give f ''(x), the second derivative.

2) Calculate f ''(a). If f ''(a)<0 then (a,b) is a local maximum.

                              If f ''(a)>0 then (a,b) is a local minimum.

To see why this works, imagine moving gradually towards our point (a,b), plotting the slope of our graph as we move. If our point is a local maximum, we can that this slope starts off positive, decreases to zero at the point, then becomes negative as we move through and past the point. Our slope, f '(x), is decreasing throughout this movement, so we must have that f ''(a)<0.

The exact reverse is true if (a,b) is a local minimum. Our slope is increasing through the same movement, so here we have that f ''(a)>0.

An example

Find the maxima and minima of f(x)=x3+x2.

First, we find f '(x). Using the rules of differentiation, we find f '(x)=3x2+2x.

Now let's set f '(x)=0:   3x2+2x=0

                                    x(3x+2)=0         (factorising)

                                    x=0 or x=-2/3

Substitute these values back in so that we can find our 'y-coordinates': f(0)=(0)3+(0)2=0


Hence our stationary points are (0,0) and (-2/3,4/27).

Finally, we use our test: f ''(x)=6x+2

                                       f ''(0)=2         (substituting x=0)

                                       f ''(-2/3)=-2    (substituting x=-2/3)

2>0, so (0,0) is a local minimum of f(x).

-2<0, so (-2/3,4/27) is a local maximum of f(x).

Alex M. A Level Maths tutor, A Level Further Mathematics  tutor, GCSE...

1 year ago

Answered by Alex, an A Level Maths tutor with MyTutor

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


PremiumRebecca V. A Level Maths tutor, 13 plus  Maths tutor, GCSE Maths tuto...
View profile
£20 /hr

Rebecca V.

Degree: MA Logic and Philosophy of Mathematics (Masters) - Bristol University

Subjects offered: Maths, Physics+ 1 more

Further Mathematics

“Hi, I'm Rebecca, I'm 23 and I'm currently taking a year to myself just to study, doing my masters in Logic and Philosophy of Maths. To me, mathematics has always been black and white, right or wrong whilst philosophy is all about thin...”

Quentin D. IB French tutor, 13 plus  French tutor, A Level French tut...
View profile
£20 /hr

Quentin D.

Degree: mathematics (Bachelors) - Durham University

Subjects offered: Maths, French


“About me: I am a French native speaker. During my childhood I have travelled quite a lot meaning that I am perfectly bilingual in French and English. Because I have travelled a lot, Maths has always been the "language" I recognized t...”

MyTutor guarantee

Hayden M. A Level Further Mathematics  tutor, GCSE Maths tutor, A Lev...
View profile
£22 /hr

Hayden M.

Degree: Physics (Masters) - Warwick University

Subjects offered: Maths, Science+ 4 more

Further Mathematics

“My name is Hayden. I am a 2nd year Physics student at The University of Warwick. I get excited aboutall things science and I hope that I can help you not only learn your material but alsounderstand why it is a useful thing to know.A...”

About the author

Alex M. A Level Maths tutor, A Level Further Mathematics  tutor, GCSE...
View profile

Alex M.

Currently unavailable:

Degree: Mathematics (Masters) - Durham University

Subjects offered: Maths, Further Mathematics

Further Mathematics

“Third year mathematics undergraduate at Durham University with previous tutoring experience. Eager to spread my enthusiasm and understanding with those studying maths at GCSE and A-level.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

Find the integral between 4 and 1 of x^(3/2)-1 with respect to x

The points P (2,3.6) and Q(2.2,2.4) lie on the curve y=f(x) . Use P and Q to estimate the gradient of the curve at the point where x=2 .

Why bother with learning calculus?

How do you find the gradient of a parametric equation at a certain point?

View A Level Maths tutors


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