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# Finding Maxima and Minima of a Function

Differentiation has many applications in our day to day lives, and one of them is in finding maxima and minima of a given function, i.e. maximum and minimum values of a function.

This has many applications in real life, like finding the maximum volume of a given shape, etc.

Let’s look at an example of finding maxima and minima of a function:

Example 1: A rectangular prism with a square base is to have a surface area of 300 cm2.

a) Show that the volume of the prism is given by V =

b) Find the dimensions of the prism that will have the maximum volume.

a) Given surface area of prism = 300 cm2,

SA y  = 2x2 + 4xy  =  300. Hence

y = x2 + 2xy  =  150, or

y =

Volume of the prism V = x2y

V = x2

=

=

b) For the volume to be maximum

= 0   and     < 0

–

= 75 –  = 0

75 =

x2

x =

Since x cannot be , x is

=   = -3x

When x = ,     =   < 0 (maximum point)

Maximum volume V =

cm3  =   cm3

Substituting the value of x =  in y = , we get

y =

y =

=

=

Maximum volume of V =  cm3 has dimensions of the prism to be x =  cm and y = cm.

In other words, maximum volume of a prism is achieved when it is a cube.

Here are some more applications of the maxima and minima of a function.