Use optimilization techniques to answer the question?

Question

Find two real numbers x and y whose sum is 36 and whose product is as small as possible.

Answer 1

Two real numbers are x,y

Sum of real numbers is x + y = 36

Step 1:  Find required form

x + y = 36 ⇒ y = 36 - x

Consider z = xy

z = xy = x(36-x)

z = 36x - x²

Step 2:

Apply derivative on both sides.Then set it to zero for determining minimum/maximum values.

dz/dx = z' = 36 - 2x

set dz/dx = 0

36 - 2x = 0

36 = 2x

x = 36/2

x = 18

y = 36 - x = 36 - 18 = 18

Step 3: Verification of solution is maximum or minimum

Apply again derivative on both sides

d²z/dx² = z'' = - 2

z'' = -2 < 0.

So solution x = 18 , y = 18 is gives maximum product value.

By using optimization technique we can obtain maximum product only in this case.