I would like to find a solution to the following problem:
I have the following utility function:
\begin{equation} U(x,r)=\alpha\frac{x^{1-r}-1}{1-r}-(1-\alpha)(x-0.5)^2, \end{equation} subject to $ x\in [0,1], \alpha\in [0,1], r\neq1 .$
I want to maximize the function w.r.t. $x$.
The first order derivative is the following:
\begin{equation} U^{'}_{x}(x,r)=\alpha x^{-r} -2x(1-\alpha)+(1-\alpha). \end{equation}
I can not find the solution to the equation
\begin{equation} U^{'}_{x}(x,r)=0 \end{equation}
In the end, if there is no analytical solution, I would like to know the sensitivity of the optimal solution w.r.t. to changes in $r$.