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Math Help - Prove pi^e < e^pi

  1. #1
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    Prove pi^e < e^pi

    Consider the function f(x)= ln(x)/x which is defined for all x>0.

    Show that this functions is strictly increasing on the interval (0,e) [ This part is done just find f'(x) and state its increasing for all x between 0 and e]

    strictly decreasing on the interval (e,infinity) [ same as the above part..]

    and thus has a global maximum at x=e ... [ DONE, proving its station point is at that point].

    HENCE SHOW THAT F(X) <= 1/e FOR ALL x>0.

    USE THIS RESULT WITH x=pi TO SHOW THAT pi^e < e^pi.....

    just that last part i dont even know where to start.....
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  2. #2
    ynj
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    Quote Originally Posted by Khonics89 View Post
    Consider the function f(x)= ln(x)/x which is defined for all x>0.

    Show that this functions is strictly increasing on the interval (0,e) [ This part is done just find f'(x) and state its increasing for all x between 0 and e]

    strictly decreasing on the interval (e,infinity) [ same as the above part..]

    and thus has a global maximum at x=e ... [ DONE, proving its station point is at that point].

    HENCE SHOW THAT F(X) <= 1/e FOR ALL x>0.

    USE THIS RESULT WITH x=pi TO SHOW THAT pi^e < e^pi.....

    just that last part i dont even know where to start.....
    a^b<b^a\Leftrightarrow \ln a^b<\ln b^a\Leftrightarrow b\ln a<a\ln b\Leftrightarrow \frac{\ln b}{b}>\frac{\ln a}{a}
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  3. #3
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    Hmmm ^^^^^^^^

    where do I go ??
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  4. #4
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    Quote Originally Posted by Khonics89 View Post
    Consider the function f(x)= ln(x)/x which is defined for all x>0.

    Show that this functions is strictly increasing on the interval (0,e) [ This part is done just find f'(x) and state its increasing for all x between 0 and e]

    strictly decreasing on the interval (e,infinity) [ same as the above part..]

    and thus has a global maximum at x=e ... [ DONE, proving its station point is at that point].

    HENCE SHOW THAT F(X) <= 1/e FOR ALL x>0.

    USE THIS RESULT WITH x=pi TO SHOW THAT pi^e < e^pi.....

    just that last part i dont even know where to start.....
    f(\pi) = \frac{\ln(\pi)}{\pi} < \frac{1}{e}

    \ln(\pi) < \frac{\pi}{e}

    e\ln(\pi) < \pi

    \ln(\pi^e) < \pi

    e^{\ln(\pi^e)} < e^{\pi}

    \pi^e < e^{\pi}
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