`x+y = 9`

`y = 9-x`

Let us say;

`A = xy`

`A = x(9-x)`

`A = 9x-x^2`

We can use the derivation to find the maximum or minimum of A. At maximum or minimums the derivative of A which is `(dA)/dx` will be zero.If A has a maximum then `(d^2A)/dx^2 <0`

`A = 9x-x^2`

`(dA)/dx = 9-2x`

When `(dA)/dx = 0` ;

`9-2x = 0`

`x = 4.5`

`(dA)/dx = 9-2x`

`(d^2A)/dx = 0-2`

`(d^2A)/dx= -2`

`(d^2A)/dx < 0`

*So A has a maximum at x = 4.5 .This means the product of x and y will be maximum when x = y = 4.5*