x+y=8 ⇒ y=8-x

xy=x(8-x)

Let f(x)=8x-x²

Differentiating f(x) with respect to x, we get

f’(x)=8-2x

Differentiating f’(x) with respect to x, we get

f’’(x)=-2<0

For maxima at x=c, f’(c)=0 and f’’(c)<0

f’(x)=0 ⇒ x=4

Also f’’(4)=-2<0

Hence, x=4 is a point of maxima for f(x) and f(4)=16 is the maximum value of f(x).