Maximizing product of fractions

Question

I want to maximize $f(a,b,c) = \frac{a}{a+1}\cdot\frac{b}{b+1}\cdot\frac{c}{c+1}$, where $a,b,c$ are nonnegative integers and $a+b+c=30$. I began the problem with AM-GM to find an upper bound of 1000 for abc, but am unsure how to proceed.

Answer 1

Writing $x,y,z$ instead of $a,b,c$ and setting the problem in 3d space, your function represents a surface which has a 3-fold rotational symmetry with respect to the line given parametrically by \begin{equation} x=t, y=t, z=t \end{equation} i.e. the "center" line of the first octant in 3d space. Your constraint, $x+y+z=30$ or more generally $x+y+z=k$ is a plane orthogonal to that line. From these considerations it follows that the solution you are searching lies at the point of tangency between your surface (the function you want to maximize) and the plane, in fact the surface \begin{equation} \frac{x}{x+1}\cdot\frac{y}{y+1}\cdot\frac{z}{z+1}=k \end{equation} shifts away from the origin as $k$ increases. To see this, try to sketch the similar function in the plane \begin{equation} \frac{x}{x+1}\cdot\frac{y}{y+1}=k \end{equation} and you will see that the solution of your original problem must be such that \begin{equation} x=y=z \end{equation}