# Optimization based on partial derivative

I want to maximize $f(a,b,c)$ over the variables $a,b,c$, such that

1) $0\leq a\leq K$,

2) $0\leq b\leq K$,

3) $0\leq c \leq K$

where $K$ is a constant.

The $f(a,b,c)$ is some rational function (2nd order polynomial divided by 1st order polynomial in $a,b,c$).

The observation is that $\frac{\partial f(a,b,c)}{\partial a} > 0$, and that $\frac{\partial f(a,b,c)}{\partial b} > 0$, for the domain. Both of these derivatives are continuous, smooth, and well-behavior on that domain.

I am wondering to solve this maximization problem, am I safe to simply put $a=K, b=K$, and simplify it to:

$Maixmize$ $f(a=K, b=K, c)$ such that $0\leq c \leq K$

Is this correct?? In this case, this would be a single variable optimization problem.

Long answer: the optimization over $a,b,c$ can be decoupled as $$\max_{0\le c\le K}\max_{0\le b\le K}\max_{0\le a\le K}f(a,b,c)$$ where the inner maximization over $a$ is done for a fixed $b,c$ etc. From your first derivative condition, if it holds for all $b,c$ in the interval, it follows that the maximum is attained at $a=K$ for all $b,c$. Similarly for $b$ (if the partial derivative is positive for all $c$ in the interval and $a=K$).