Maximization with inequality constraints

Question

I have to solve this optimization problem

$$\underset{x}{\max} \left(AB-\frac{xC}{D}(E+F)\right)$$ subject to
$$ \frac{AB}{x}-\frac{C}{D}(E+F) \leq G $$ and $$0 < x \leq \frac{ABD}{C(E+F)}. $$

How can I solve it?

Answer 1

Hint: The first constraint implies $AB-xC(E+F)/D \le Gx$. If $G>0$ then the maximum value of $x$ achieves the maximum.

Answer 2

Provided $A,B,C,D,E,F,G$ are constant, then: $$AB-\frac{xC}{D}(E+F)\le Gx\le \frac{ABDG}{C(E+F)}.$$