Consider the graph of the parabola \(y=2x-x^2\) and the line \(y=x\) on the interval \([0,1]\). A rectangle \(PQRS\), with sides parallel to the axes, is constructed so that the two left corners lie on the \(y\)-axis, the upper right corner lies on the parabola \(y=2x-x^2\), and the lower right corner lies on the line \(y=x\).

In terms of \(x\), the points \(Q\) and \(R\) are

\[ Q = \left (x,\answer {x}\right )\text { and } R = \left (x,\answer {2x-x^2} \right ). \]

The height of the rectangle \(PQRS\), in terms of \(x\), is

\[ h(x) = \answer {x-x^2}. \]

The area of the rectangle \(PQRS\), in terms of \(x\), is

\[ A(x) = \answer {x^2-x^3}, \]
and the domain of this function is
\[ \left [\answer {0},\answer {1}\right ]. \]

The value of \(x\) that maximizes the area of \(PQRS\) is

\[ x = \answer {\frac {2}{3}}. \]