Hello. I am really stuck with this problem. Please help.

x and y are positive numbers. Which of the following implies $\displaystyle x^y \ge y^x$

A: $\displaystyle x \leq e \leq y$

B: $\displaystyle y \leq e \leq x$

C: $\displaystyle x \leq y \leq e \text { or } e \leq y \leq x$

D: $\displaystyle y \leq x \leq e \text{ or } e \leq x \leq y$

The answer should be D, but I don't know how.