The pair of numbers whose sum is 8 and whose product is as large as possible is

X and y are the numbers

x+y=8

and product is max

xy=max

ok

x+y=8

minus x both sides

y=8-x

subsitute 8-x for y in other equation

(8-x) (x) = max

8x-x²=max

take derivitive

8-2x=dy/dx max

find where it equal 0

8-2x=0

8=2x

4=x

at x=4

y=8-x

y=8-4

y=4

the numbers are 4 and 4

their product is 4*4 or 16