How to prove
$$ \Bigl(\dfrac 1n\Bigr)^n + \Bigl(\frac 2n\Bigr)^n + \cdots + \Bigl(\frac nn\Bigr)^n \geqslant \frac{3n+1}{2n+2} \qquad (n\in\mathbb{N}) $$
I tried:
let $f(x)=x^n$ and $f''(x)\geqslant 0$ for $n>1$, let $x_i=\frac in$, and we have
$$ \sum f(x_i)\geqslant nf\biggl(\frac{\sum x_i}{n}\biggr). $$
But the $RHS<\dfrac{3n+1}{2n+2}$. So it doesn't work.
Could someone give me a neat proof? Thanks!