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?