Inequality

**doxian** · April 12th 2010, 12:56 PM

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

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

A: $x \leq e \leq y$
B: $y \leq e \leq x$
C: $x \leq y \leq e \text { or } e \leq y \leq x$
D: $y \leq x \leq e \text{ or } e \leq x \leq y$

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

**mathemagister** · April 12th 2010, 11:15 PM

Originally Posted by doxian

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

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

A: $x \leq e \leq y$
B: $y \leq e \leq x$
C: $x \leq y \leq e \text { or } e \leq y \leq x$
D: $y \leq x \leq e \text{ or } e \leq x \leq y$

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

$x^y \ge y^x$

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

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

$\frac{\ln{n}}{n}$ is decreasing when when $n > e$ and increasing when $n < e$ (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

**doxian** · April 12th 2010, 11:43 PM

Thanks a lot for your nice answer!

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