Is it possible to find the number of turning points of a power function whose largest exponent is some real number known to lie between $(n,n+1)$ for some $n\in\mathbb{Z}$?

To give an example

Consider the function $f:(0,1)\rightarrow\mathbb{R}$ with: $$ f(z)=A(1-z)^{\gamma+1}+Bz(1-z)^{\gamma}+Cz^{\gamma+1} $$

where A,B,C are real constants and $\gamma\in(0,1)$. This implies that the highest power of $z$ lies in (1,2), and my initial intuition was that this should imply that the function has at most 1 turning point. Any suggetsions on how to go about proving this?