Question

Of all the numbers whose difference is 54, find the two that have the minimum product.

Answer 1

Let's suppose those numbers are x and y

given x-y = 54

So, x = 54+y

Now the Product of these numbers is

y(54+y) = y^2+54y .

Our task is to minimise the value of y^2+54y and for this task we will take the help from differentiation.

step 1: find the critical point of y^2+54y

for critical point

So, 2y+54=0

Or, y = -27

step2: by 2nd differentiation test let's find whether y = -27 is minima or maxima

Here 2nd derivative is positive so y^2+54y have monima at y=-27.

So value of y=-27 and value of x is 54+y

So, x = 54+(-27) = 27

The two numbers are 27 and -27

and their minimal product is -729.