We impose that the first number =X

the second =y

the maximum product=m(x)

y+x=72

y=72-x

M(x)=x * y

M(x)=x *(72-x)

M(x)=72x- X^2 (by deriving the both sides of the equation)

M'(x)=72-2X

we Put M' (x)=0

72 - 2X=0

2X =72

X= 36 Y =72 -36

Y =36

the second =y

the maximum product=m(x)

y+x=72

y=72-x

M(x)=x * y

M(x)=x *(72-x)

M(x)=72x- X^2 (by deriving the both sides of the equation)

M'(x)=72-2X

we Put M' (x)=0

72 - 2X=0

2X =72

X= 36 Y =72 -36

Y =36