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

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.