Finding the maximal efficiency

We want to calculate the maximum value of windmill efficiency $P/P_0$. Write $y$ for $P/P_0$ and $x$ for $V/U$. We get $$y=\frac{1}{2}(1-x^2)(1+x).$$ Differentiating we get $$dy/dx=\frac{1}{2}(1-2x-3x^2).$$ The maximum occurs when $dy/dx=0$, in other words when $$3x^2+2x-1=0.$$ This happens for $x_0=1/3$ and $x_1=-1$. Discounting the negative solution we get a maximal efficiency of $$1/2(1-x_0^2)(1+x_0)=16/27.$$