Maximizing a monotonically increasing function

Question

Let $f(x_1,\dots,x_n)$ be a monotonically increasing function in each of its variables. For instance, this means that for a given $(x_2,\dots,x_n)$, the one dimensional function $f(x_1,\dots)$ will be a monotonically increasing function in $x_1$. Let $g(x_1,\dots,x_n)$ be also a monotonically increasing function in each of its variables. Let $B$ be a positive constant. Also let us say that $f(.)$ and $g(.)$ are non-negative functions as well. You can assume continuity and differentiability as well. In addition, assume that there exists at least one point such that $g(x_1,\dots,x_n)=B$. What can we say about the optimization problem \begin{align}\max_{x_1,\dots,x_n}~&~f(x_1,\dots,x_n) \\s.t.&~~g(x_1,\dots,x_n)\leq B\\& ~~x_i\geq 0~,~\forall i\end{align}I am interested to know if there is a general approach to this type of problems.