Optimal cut

For a position $x=x_0$ for the dividing linie we can caculate the orange area $A$ by the integrals $A(x_0) = \int_0^{x_0} f(x) dx + \int_{x_0}^1 (1 - f(x)) dx$ We zero the derivative to find the minimum $A(x_0) \stackrel{!}{=} \text{min}$ : $A'(x_0) = f(x_0) - (1 - f(x_0) ) = 2 f(x_0) - 1 \stackrel{!}{=} 0 \quad \Rightarrow \quad f(x_0) = \frac{1}{2}$ To check whether there is actually a minimum, we also calculate the second derivative $A''(x_0) = 2 f'(x_0) > 0$ since the function is monotonic increasing. Therefore we found a single mimimum for $f(x_0) = \frac{1}{2}$ (point $C$ ).