Optimizing a function of two variables under constraint

Question

Consider the following function $f(x,y)$ :

$$f(x,y)=\frac{c_1}{c_2x+c_3y}+ \frac{c_4}{c_5-c_6x-c_7y} + c_8$$

Where $c_1,c_2,c_3,c_4,c_5,c_6,c_7,c_8>0$.

The optimization problem is:

minimize $f(x,y)$

s.t $x>0\\ y>0$

After solving $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$, I got

$y=-\frac{c_2x}{c_3}$

$x=\frac{c_3c_5}{c_3c_6-c_2c_7}$

By replacing $x$,

$y=-\frac{c_2c_5}{c_3c_6-c_2c_7}$

Now, from the expression, both $x$ and $y$ can not be positive at the same time and these values make the denominator of the first term $0$. How to solve this problem?

Answer 1

There's no mimimum.

For $x,y > 0$, $f$ is unbounded below.

Just choose $\epsilon$ positive and sufficiently small, and take $x,y > 0$ on the line $c_6x+c_7y=c_5 + \epsilon$.