# Prove that $(1+x)^k/k + (1-x)^m/m\geq 1/k +1/m$ without calculus

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.

• You should post that "real statement". This one can be written as $\,(y + x^3 - 3 x^2 + 3 x)^2 + 12 x^2 + 3 \ge 3\,$ which of course proves it, but wouldn't necessarily work for your other, untold question.
– dxiv
May 13, 2022 at 5:17
• I have made it somewhat more general now (although, I also changed it to a single variable statement). For now, I want to avoid writing the full statement because it has a lot of parameters which obfuscate what's going on. However, if this statement is still too simple, I will add more to it! May 13, 2022 at 16:39
• If $k=1$ and $m=3$, it tends to $-\infty$ when $x\to + \infty$. May 13, 2022 at 16:45
• Sorry. Should have had constraints on x. Added now. May 13, 2022 at 16:48