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I Π^x and x^π

  1. Jun 26, 2017 #1
    Guys, could you help me how to solve the inequality π^x - x^π < 0??
     
  2. jcsd
  3. Jun 26, 2017 #2

    jedishrfu

    Staff: Mentor

    The easiest way to solve this is to use the desmos graphing calculator site:

    https://www.desmos.com/calculator

    and type in: pi^x - x^pi

    It will show you a plot of the curve from which you can see where the zeros are and where the <0 segment is.
     
  4. Jun 26, 2017 #3
    but if we're not allowed to use any calculator?? because my school doesn't allow us to use calculator for most of my math lesson
     
  5. Jun 26, 2017 #4

    jedishrfu

    Staff: Mentor

    Okay, but since you've posted it, you could look at the graph and then see if you can devise a strategy to solve it.

    One obvious solution is: ##\pi^\pi - \pi^\pi## which is one of its zeros.

    Next, what math course is this for?

    Can you use an approximation strategy like evaluating a few terms in its Taylor series?

    Also you can try x=0, x=1... and attempt to plot it.
     
  6. Jun 26, 2017 #5
    yeah, at first i think the solution is x<π. But when i look at the graph, there are another solution that make the inequalities become smaller than zero. Anyway i got this question from my math textbook where i study by myself. and yes i can use a bit of approximation of taylor series
     
  7. Jun 26, 2017 #6

    Mark44

    Staff: Mentor

    Please post textbook problems in the Homework & Coursework sections, not here in the technical math sections.
     
  8. Jun 26, 2017 #7
    Function ##f(x)=π^x - x^π## is continuous. Find values of x when f(x)=0 aka ##π^x - x^π=0##. Ranges where f(x)<0 aka ##π^x - x^π<0## must be between those x values, in range between -∞ and smallest such x value or in range between biggest such x value and ∞.
     
  9. Jun 28, 2017 #8

    lavinia

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    Science Advisor

    This is a standard Calculus 1 homework problem
     
  10. Jun 28, 2017 #9
    Really? I haven't heard of Lambert W function until well out of university. But then it wasn't a mathematical university.
     
  11. Jun 29, 2017 #10

    lavinia

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    Science Advisor

    I never heard of the Lambert W function.
     
    Last edited: Jun 29, 2017
  12. Jun 29, 2017 #11
    Wolframalpha gives the solution in terms of the LambertW function. Is there an easier expression for the 2.3821790879930187746?
     
  13. Jun 29, 2017 #12

    lavinia

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    Science Advisor

    I don't know what this link tells you.

    I think you want to solve ##log(x)/x > log(π)/π## since

    ##π^{x} - x^{π} <0 ⇒ e^{xlog(π)} < e^{πlog(x)} ⇒ xlog{π} < πlog{x}##
     
  14. Jun 29, 2017 #13
    So how do you solve that using Calculus 1 knowledge?
     
  15. Aug 10, 2017 at 11:55 AM #14

    Svein

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    Science Advisor

    Start with observing that [itex] \frac{\log(\pi)}{\pi}[/itex] is a constant.
     
  16. Aug 13, 2017 at 8:23 PM #15
    And continue how? Remember this is not a proof of existence, we're looking for the value of x where ##\log x/x=\log\pi/\pi##.
     
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