1. ## Inequality

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.

2. Originally Posted by doxian

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.
$\displaystyle x^y \ge y^x$

Take the logarithm and use log laws: $\displaystyle y\ln{x} \ge x\ln{y}$

$\displaystyle \frac{\ln{x}}{x} \ge \frac{\ln{y}}{y}$

$\displaystyle \frac{\ln{n}}{n}$ is decreasing when when $\displaystyle n > e$ and increasing when $\displaystyle n < e$ (which you can calculate using calculus, but since this is precalc, I guess you would have to use a graph).

Do you see why it has to be D now?

Hope that helped

Mathemagister