Answer
$$27{\text{ and }}54$$
Work Step by Step
$$\eqalign{
& {\text{Let }}x{\text{ and }}y{\text{ be the numbers}} \cr
& {\text{We have that:}} \cr
& x + 2y = 108{\text{ }}\left( {\bf{1}} \right) \cr
& {\text{The product of the numbers}} \cr
& P = xy{\text{ }}\left( {\bf{2}} \right) \cr
& {\text{Solve the equation }}\left( {\bf{1}} \right){\text{ for }}y \cr
& y = \frac{{108 - x}}{2} \cr
& {\text{Substitute }}\frac{{108 - x}}{2}{\text{ into equation }}\left( {\bf{2}} \right) \cr
& P = x\left( {\frac{{108 - x}}{2}} \right) \cr
& P = 54x - \frac{{{x^2}}}{2} \cr
& {\text{Differentiate}} \cr
& \frac{{dP}}{{dx}} = 54 - x \cr
& {\text{Find the critical points by solving }}\frac{{dP}}{{dx}} = 0 \cr
& 54 - x = 0 \cr
& x = 54 \cr
& {\text{Calculate the second derivative}} \cr
& \frac{{dP}}{{dx}} = - 1 \cr
& {\text{By the second derivative test:}} \cr
& {\left. {\frac{{{d^2}P}}{{d{x^2}}}} \right|_{x = 54}} = - 1 < 0{\text{ Relative maximum}} \cr
& {\text{Calculating }}y \cr
& y = \frac{{108 - x}}{2} \to y = \frac{{108 - 54}}{2} = 27 \cr
& {\text{Therefore, the numbers are: }}27{\text{ and }}54 \cr} $$
