Let $f(x)=x^{n+1}+a x^n$, where a > 0. Then, x = 0, is a point of

local minimum for any integer n local maximum for any integer n local minimum if n is an even integer local minimum if n is an odd integer

local minimum if n is an even integer

We have, $f(x)=x^{n+1}+a x^n$ $\Rightarrow f^{\prime}(x)=(n+1) x^n+n a x^{n-1}$ $\Rightarrow f^{\prime}(x)=[(n+1) x+a] x^{n-1}$ If n is even, then (LHD at x = 0) > 0 and (RHD at x = 0) < 0. Thus, f(x) has a local maximum at x = 0.