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Math Help - 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 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.
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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 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
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  3. #3
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    Thanks a lot for your nice answer!
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