MATH Versus MATH




A classical math riddle is to ask: "Which is bigger, $e^{\pi }$ or $\pi ^{e}$?" Of course, you're not allowed to use a calculator! Here are some hints for solving this problem.




  1. Consider the general problem: When is $a^{b}$ bigger than $b^{a}$? Suppose then that $a<b$ (e.g. $e<\pi $). Start with the inequalityMATH

    and raise both sides to the $1/b$ power, then to the $1/a$ power.

  2. The problem now is, given $a<b$, when is $a^{1/a}<b^{1/b}$. (Explain how this is related to step 1.)

  3. You must now examine the graph of $y=x^{1/x}$. A graphing calculator will be helpful; remember to plot in a window that starts a little to the right of $x=0$, since $0$ is not in the domain of this function). If you know enough calculus to differentiate and work with $x^{1/x}$ then use it!Find the following:MATH

  4. So where is the function $x^{1/x}$ increasing and where is it decreasing?

  5. Which is bigger, $e^{1/e}$ or $\pi ^{1/\pi }$?

  6. So which is bigger, $e^{\pi }$ or $\pi ^{e}$ ?