Find two positive real numbers with the given sum whose product is a maximum. The sum of the first and twice the second is 24

Y+2x=24

product is max

xy=max

we have

y+2x=24

minus 2x both sides

y=24-2x

sub that for y in xy=max

(24-2x) (x) = max

-2x^2+24x=max

we have a parabola opening down

the coordinates of the vertex is our answer

the x value of the vertex is - b/2a where

ax^2+bx+c=y

-24/2 (-2) = - 24/-4=6

xvalue=6

we can just sub

y=24-2x

y=24-2 (6)

y=24-12

y=12

the numbers are 6 and 12

the product is 72

the numbers are 6 and 12