When I was a senior in high school in 2001, as I took calculus, I made the following conjecture that proves resistive to attack. It goes like this: For every positive integer $n$, there are exactly $n$ positive real zeroes of $$\frac {d^n}{dx^n}x^{1/x}$$ and no two derivatives has a common root.
A few things to keep in mind are that it is not hard to show that $$\lim_{x\rightarrow 0^+}\frac {d^n}{dx^n}x^{1/x}=\lim_{x\rightarrow \infty}\frac {d^n}{dx^n}x^{1/x}=0$$ for all $n\geq 1$ and that $x^{1/x}$ is smooth and then to use these to show that the $n$th derivative has at least $n$ positive real zeroes. The proof of smoothness uses induction on $n$.