The curve \(C_1\) passes through the origin in the \(x\text{-}y\) plane and its gradient is given by \[ \frac{dy}{dx}=x(1-x^2)e^{-x^2}.\] Show that \(C_1\) has a minimum point at the origin and a maximum point at \((1,\frac{1}{2}e^{-1})\). Find the coordinates of the other stationary point. Give a rough sketch of \(C_1\).
The curve \(C_2\) passes through the origin in the \(x\text{-}y\) plane and its gradient is given by \[ \frac{dy}{dx}=x(1-x^2)e^{-x^3}.\] Show that \(C_2\) has a minimum point at the origin and a maximum point at \((1,k)\), where \(k>\frac{1}{2}e^{-1}\). (You need not find \(k\).)
