Let x and y be positive numbers . Which of the following always implies $x^y \geq y^x $
a.) $x\leq e\leq y$
b.)$y\leq e \leq x$
c.)$x\leq y \leq e$ or $e\leq y \leq x$
d.)$ y\leq x \leq e$ or $e\leq x \leq y $
My attempt : To solve the following inequality I tried taking log on both sides , however I could could not progress much farther than that. I tried assuming two cases after that one where both x and y are greater than e and one where both are less than e.However I could not proceed further. Please help.