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
Volume of the prism V = x2y
V = x2
b) For the volume to be maximum
= 0 and < 0
= 75 – = 0
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 = =
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.