Can anyone explain to me how the "smoothing argument for inequalities" works?
I know that basically it can be used to prove an inequality $f(a_1,a_2,\cdots,a_n)\geq C$ subject to the constraint $a_1+a_2+\cdots+a_n=k$ (for constants $C,k$) if equality holds when all $a_i$ are equal. I've heard that the basic idea is to push the $a_i$ closer together, showing that this is pushing $f(a_1,a_2,\cdots,a_n)$ to an extrema.
After doing some research, I saw that this can be rigorously defined by showing inequalities of the form $f(a_1,a_2,\cdots,a_n)\geq f(\frac{k}{n},x_1+x_2-\frac{k}{n},\cdots,x_n)$, but I don't really see a pattern here (why do you replace $a_1$ with $\frac{k}{n}$, $a_2$ with $x_1+x_2-\frac{k}{n}$, etc.) Can someone explain this?
Finally, could someone show me how to apply this to an actual inequality problem? For example, how would one solve the following inequality using smoothing (yes, I am aware that there is a quite simple solution using substitutions and AM-GM)?
Let $a_1,\cdots a_n$ be real numbers in the interval $(0,\frac{\pi}{2})$ such that $\sum_{i=0}^n \tan(a_i-\frac{\pi}{4})\geq n-1$. Prove that $\prod_{i=0}^n\tan(a_i)\geq n^{n+1}$