A tricky inequality: $n(n+1)^{\frac{1}{n}}+(n-2)n^{\frac{1}{n-2}}>2n,\ n\geq3.$

By Bernoulli $$n(n+1)^{\frac{1}{n}}+(n-2)n^{\frac{1}{n-2}}=\frac{n}{\left(1+\frac{1}{n+1}-1\right)^{\frac{1}{n}}}+\frac{n-2}{\left(1+\frac{1}{n}-1\right)^{\frac{1}{n-2}}}\geq$$ $$\geq\frac{n}{1-\frac{1}{n+1}}+\frac{n-2}{1+\frac{1}{n-2}\cdot\frac{1-n}{n}}=n+1+\frac{n(n-2)^2}{n^2-3n+1}=$$ $$=2n+\frac{1}{n^2-3n+1}>2n$$

Hint

You can use a majorization theorem as follows :

Let $a\geq b>0 $ and $c\geq d>0$ if we have :

$a\geq c $ $\, \operatorname{and}$ $ab\geq cd $ then we have :

$$a+b\geq c+d$$

Here you can take $c=d=n$ and $a=n(n+1)^{\frac{1}{n}}$,$\,$$b=(n-2)n^{\frac{1}{n-2}}$

Then it's easier to use derivative .The theorem I used is a direct consequence of the Karamata's inequality .