Will the returned result of the function
$$\max\{\tfrac{1}{a}+\tfrac{1}{f}, \tfrac{1}{b}+\tfrac{1}{e}, \tfrac{1}{c}+\tfrac{1}{d}\}$$
return the same set $\{a,f\}$, $\{b,e\}$ or $\{c,d\}$ as the function
$$\min\{a+f, b+e, c+d\}\quad?$$
Assume all numbers are positive and real-valued.
In other words, if, for example, $$a+f < b+e\quad\text{ and }\quad a+f < c+d,$$ will it be true that $$\tfrac{1}{a}+\tfrac{1}{f} > \tfrac{1}{b}+\tfrac{1}{e}\quad\text{ and }\quad \tfrac{1}{a}+\tfrac{1}{f} > \tfrac{1}{c}+\tfrac{1}{d}\quad ?$$