Question

Find two numbers whose sum is 24 and whose product is as large as possible.

Answer 1

It is given that the sum of two numbers is 24.

Consider that the first number is x, so, second number will be 24−x.

Let P( x ) be the product of two numbers, so the function becomes,

P( x )=x( 24−x )

Differentiate the function with respect to x,

P ′ ( x )=24−2x(1)

Put P ′ ( x )=0,

24−2x=0 2x=24 x=12

Differentiate equation (1) with respect to x,

P ″ ( x )=−2 <0

This shows that the function is maxima.

Therefore, the first number x is 12 and the second number, 24−x is 12.