The function $f(x) = x^x, x > 0$ can be plotted on graphing software and inspected to see a local minimum around .367. The function is convex, decreasing from 0 to its minimum, and increasing thereafter. The derivative of the function can be found by implicit differentiation to be $f'(x) = x^x(\ln(x)+1)$.
- Can the exact value of the local minimum be found?
- Can someone explain intuitively why the function first decreases and then increases?
- Is there anything interesting about the class of functions $\{ \, f(x) \, | \, f'(x) = f(x) \cdot g(x) \, \}$
Edit: I forgot, I'll be subject to a firing squad if I don't explain what I have tried!
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Setting the derivative equal to 0 doesn't do much for me. If this can be solved exactly without optimization algorithms, I suspect something really clever will need to happen.
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I tried thinking about what is happing for different types of inputs (irrational $x$, rational $x$, natural $x$). Still didn't get anywhere.
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Nothing to try really, does this class of functions come up anywhere in math?
Best Answer
The minimum is attained at $x=\frac1e$ as can be seen by setting the derivative equal to $0$ and solving for $x$.
Your statement in a comment that it is not always true that if $x>y>0$ then $x^x>y^x$ is false. We have $$x>y\implies \log x > \log x\implies x\log x > x\log y \implies x^x>y^x $$
Perhaps you meant to say it is not always true that $x>y>0$ implies $x^x>y^y$. This is true. Taking logarithms, there's no particular reason to believe that $x\log x > y\log y$ and indeed, it isn't always true.
To try to explain it intuitively, note first that we're concerned with numbers $0<x<y<1$. When you raise such a number to a power, the larger the power, the smaller the result. When we compare $x^x$ and $y^y$, for some choices of $x$ and $y$, this effect dominates, and $y^y$ is the smaller of the two numbers. For other choices of $x$ and $y$, the fact that $x$ is smaller dominates.