How to prove that a polynomial function has single extermum in an interval?

Question

I have a function in the form of:

$$F(x)=\sum_{i=1}^N x^{2a_i}-2\sum_{i=1}^k x^{a_i},$$

where $k<N$, $\forall i: a_i \in \mathbb{R}, a_i>0$.

I need to prove that this function has at most one (or maybe exactly one) critical point in the interval of $(0,1)$. Could you please help me with that?