MYTUTOR SUBJECT ANSWERS

746 views

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

                                                                                                                 f(-2/3)=(-2/3)3+(-2/3)2=4/27

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...

2 years ago

Answered by Alex, an A Level Maths tutor with MyTutor


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

294 SUBJECT SPECIALISTS

£20 /hr

Matthew H.

Degree: Computer Science and Maths (Masters) - Exeter University

Subjects offered:Maths, Further Mathematics + 1 more

Maths
Further Mathematics
Computing

“I'm a Computer Science & Maths student at Uni of Exeter, I am a kind, patient and willing tutor who pushes my students to achieve something each tutorial.”

MyTutor guarantee

£20 /hr

Sam T.

Degree: Mathematical Physics (Bachelors) - Edinburgh University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
Chemistry

“I’m studying Physics with units in Maths and Chemistry. I really enjoy helping other people find a love for those subjects even if they think they're the worst.”

£24 /hr

Prina S.

Degree: Environmental Geoscience (Masters) - Bristol University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Geography
Biology

“Hello there! My name is Prina and I'm a first year Environmental Geoscience at the University of Bristol. My love for science was instilled in me by those who taught me and I hope to share my passion in the same way through exciting an...”

About the author

Alex M.

Currently unavailable:

Degree: Mathematics (Masters) - Durham University

Subjects offered:Maths, Further Mathematics

Maths
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

How to differentiate using the Product Rule

Differentiate y=x(e^x)

Find the coordinates of the minimum point of the curve y = 3x^(2) + 9x + 10

Differentiate y=x*ln(x^3-5)

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

mtw:mercury1:status:ok