(a) A Pringle crisp can be defined as the surface

$$z=x y \quad \text { with } \quad x^{2}+y^{2} \leqslant 1$$

Use the method of Lagrange multipliers to find the minimum and maximum values of $z$ on the boundary of the Pringle crisp and the $(x, y)$ positions where these occur.

(b) A farmer wishes to construct a grain silo in the form of a hollow vertical cylinder of radius $r$ and height $h$ with a hollow hemispherical cap of radius $r$ on top of the cylinder. The walls of the cylinder cost $£ x$ per unit area to construct and the surface of the cap costs $£ 2 x$ per unit area to construct. Given that a total volume $V$ is desired for the silo, what values of $r$ and $h$ should be chosen to minimise the cost?