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 \(C_1\).

What condition is required for a stationary point, and how do we classify them?

To find the coordinates of the maxima, we need to know \(y\) as a function of \(x\). What form do you think the function will take?

Other things to consider when sketching this function:

- what happens as \(x \rightarrow \pm\infty\)?
- is this function even or odd (or neither)?

Show that \(C_2\) has a minimum point at the origin and a maximum point at \((1,k)\), where \(k>\frac{1}{2}e^{-1}\).

Can we find the \(x\) coordinates of the stationary points? Can we classify them?

Can we prove the inequality involving \(k\) by comparing with the function \(C_1\)?