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

Explanation

Let first number be x , then second number is (24-x).
`:.` Product of two numbers
` y = x(24-x) = 24x - x^(2)`
` rArr (dy)/(dx) = 24 - 2x and (d^(2)y)/(dx^(2) =- 2`
Now, put `(dy)/(dx) = 0`,
`rArr 24 - 2x = 0 rArr x = 12`
at ` x= 12, (d^(2)y)/(dx^(2)) =- 2 lt 0` ltbregt ` rArr` y is maximum at x = 12
` :. ` second number = 24 - 12 = 12
Therefore, the numbers are 12 and 12.