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Q: What will the graph of $f(x)=(\frac{x!}{x^x})^{1/x}$ look like, and what is the method of drawing it(by hand)?

I can draw graphs of simple polar and parametric functions which include finding the increasing or decreasing part of the function, its maxima minima, its concave and convex parts, point of inflection etc.

However, I do not know how to plot $x!$ or $x^x$.

So unfortunately this whole function $f(x)=(\frac{x!}{x^x})^{1/x}$ is really difficult for me to plot.

I need help solving it!

Thank you.

Ѕᴀᴀᴅ
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  • What is $3.2!$ ? – Jack M Sep 25 '18 at 23:06
  • Forget about plotting it - do you even know how to calculate it? Because I don't. How do you multiply "the first $3.2$ numbers" together? It sounds like what you're proposing is to round down $x$ to the next lowest integer before applying the factorial? – Jack M Sep 25 '18 at 23:23
  • As far as i know factorials can be calculated for every x>0. If x is a non-integer it can still be calculated as it corresponds to a form of the Gamma function. (https://math.stackexchange.com/questions/1667023/factorial-calculation-for-non-integers(this link contains auch an example solved in mathstack) –  Sep 25 '18 at 23:36
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    Okay then. To answer your original question, I think the answer is probably pretty much "yup, it's hard to graph". Taking derivatives to work out at least where the function is increasing and decreasing may help, but then you'd need to derive the gamma function. – Jack M Sep 25 '18 at 23:38
  • Yes.That is really difficult. Thanks for the help. –  Sep 25 '18 at 23:41

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