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Thread: Inequality

  1. #1
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    Inequality

    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.
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  2. #2
    Member mathemagister's Avatar
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    Quote Originally Posted by doxian View Post
    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.
    $\displaystyle x^y \ge y^x$

    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
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  3. #3
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    Thanks a lot for your nice answer!
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