Is minimization of a linear function is equivalent to maximization of its inverse?

Question

Let us consider a linear programming problem shown below: \begin{align} \min_x f(x)-c \end{align} where, $f(x)$ is a linear function of $x$. Now, can be following maximization problem be considered as an equivalent to the above minimization problem? \begin{align} \max_x \left(\frac{1}{f(x)-c}\right) \end{align}

Answer 1

It seems to work only if $$f(x)-c> 0 \quad \lor \quad f(x)-c< 0 $$

Indeed if $f(x)$ is increasing, that is $f'(x)>0$, we have that $\frac{1}{f(x)-c}$ is decreasing, since

$$\left(\frac{1}{f(x)-c}\right)'=\frac{-f'(x)}{(f(x)-c)^2}<0$$

and if $f(x)$ is decreasing that is $f'(x)>0$ we have that $\frac{1}{f(x)-c}$ is increasing.