Maximize $f(x, y)$ subject to a linear inequality constraint

Question

For a function $f(x, y)$ we have:

1) A linear inequality constraint like: $x + 5y\leq 20$

2) $\frac{df}{dy}\geq0$

Can we conclude that in order to maximize $f(x, y)$ the equality must hold ? (here $x + 5y = 20$)

Answer 1

"Must" hold, no. But "can assume that it holds", yes.

Because ${\partial f\over \partial y} \ge 0$, $f$ is increasing in the $y$-direction. That is, for all $x, y_1, y_2$ with $y_1 \le y_2$, we have $f(x, y_1) \le f(x, y_2)$. Now for any $(x, y)$ with $x + 5y < 20$, we have $y < \frac{20 -x}5$, so $f(x, y) \le f\left(x, \frac{20 -x}5\right)$

Thus, if $f$ has a maximum value anywhere in the region, it must also take on that same value on the borderline directly above the maximum point. So if you are trying to find the maximum value, it is sufficient to look only at the border. But if you are trying to find all $(x, y)$ at which the maximum is obtained, it is not sufficient. $f$ could also obtain the maximum off the border, although in that case $f$ will be constant on the line from the maximum point up to the border.