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

Explanation

Let first number be `x`
Now, given that
First number +Second number `=24`
`x`+Second number `=24`
Second number =`24-x`
Product=(first number)`times `(second number)
`=x(24-x)`
Let `P(x)=x(24-x)`
We need product as large as possible
Hence we need to find maximum value of `P(x)`
Finding `P'(x)`
`P(x)=x(24-x)`
`P(x)=24x-x^2`
`P'(x)=24-2x`
`P'(x)=2(12-x)`
Putting `P'(x)=0`
`2(12-x )=0`
`12-x=0`
`x=12`
Finding `P''(x)`
`P'(x)=24-2x`
`P''(x)=0-2`
`=-2`
Thus, `P''(x)<0` for `x=12`
`x=12`is a point of maxima &`P(x)` is maximum at `x=12`
Finding maximum `P(x)`
`p(x)=x(24-x) `
Putting `x=12`
`p(12)=12(24-12)=12(2)=144`
First number `=x=12` &Second number `=24-x=(24-12)=12`