Let's assume there are two functions $f(x)$ and $g(x)$. I want to know when the optimal $x$ of max of min of $f(x)$ and $g(x)$ is not equal to optimal $x$ of min of max of $\frac{1}{f(x)}$ and $\frac{1}{g(x)}$ i.e. when (under which condition) the following expressions yield different optimal $x$ ?
$$\max_x \min\left(f(x),g(x)\right)$$ and
$$\min_x \max\left(\frac{1}{f(x)},\frac{1}{g(x)}\right)$$