lean_workbook_plus_16937

Let $f(x) = 2x(1-x)^2$ , then the max occurs when $f'(x) = 6x^2 - 8x + 2 = 0$ , which has roots $x = \frac{1}{3}$ and $x=1$ . Hence, \n\n $f(x) \leq f(\frac{1}{3}) = 2(\frac{1}{3})(\frac{2}{3})^2 = \frac{8}{27}$ .