The sum of two number is constant. Show that their product will be maximum if each number is half of their sum.

Explanation

Let the numbers be x and y and their sum 'a' is constant.
`:. X+y = a` …(1)
Let the product of number be P.
`:. P= x* y`
`=x(a-x)` [From (1)]
`=ax - x^(2)`
`rArr (dP)/(dx) = a - 2x`
For maxima/minima
`(dP)/(dx) = 0`
`rArr a-2x = 0`
`rArr x = a//2`
and `(d^(2)P)/(dx^(2)) = - 2 lt 0`
`rArr" P is maximum at " x= a//2`
Now `y=a-x=a-a//2=a//2`
Therefore, for the maximum product, each number will be half of their sum.