Maximize
$$f(x) = \left(x^2 + x \left(\frac{N}{P_2} - \frac{P_1}{P_2} x \right) + \left(\frac{N}{P_2} - \frac{P_1}{P_2} x \right)^2 \right)^\alpha$$
where $x \in [0,\frac{N}{P_1}]$. Moreover $P_1, P_2>0$, $\alpha \in (0,1)$ and $N>0$.
I think maximizing $f$ is the same as maximizing $$g(x)=(x^2+x(N/P_2-P_1/P_2 x)+(N/P_2-P_1/P_2 x)^2)$$ critical points are the endpoints and points where f'=0, but I do not understand whether this critical points are in the set $x \in [0,N/P_1]$ also to study the sign of f'. Can you help me?