Note: this has been edited to make the question more general.
I want to show that $(1+x)^k/k + (1-x)^m/m$ is minimized at $x=0$ when $k,m\geq 1$ and $-1\leq x \leq1$.
Of course, I could take the first two derivatives, etc. However, the real statement that I need to prove is a more general multivariate version of this, and it is very ugly to take the gradient and Hessian of it in full generality.
What I'm really hoping is that there's some simple trick using Hölder's Inequality, or something related to it.