(a) A surface in $\mathbb{R}^{3}$ is defined by the equation $f(x, y, z)=c$, where $c$ is a constant. Show that the partial derivatives on this surface satisfy

$$\left.\left.\left.\frac{\partial x}{\partial y}\right|_{z} \frac{\partial y}{\partial z}\right|_{x} \frac{\partial z}{\partial x}\right|_{y}=-1$$

(b) Now let $f(x, y, z)=x^{2}-y^{4}+2 a y^{2}+z^{2}$, where $a$ is a constant.

(i) Find expressions for the three partial derivatives $\left.\frac{\partial x}{\partial y}\right|_{z},\left.\frac{\partial y}{\partial z}\right|_{x}$ and $\left.\frac{\partial z}{\partial x}\right|_{y}$ on the surface $f(x, y, z)=c$, and verify the identity $(*)$.

(ii) Find the rate of change of $f$ in the radial direction at the point $(x, 0, z)$.

(iii) Find and classify the stationary points of $f$.

(iv) Sketch contour plots of $f$ in the $(x, y)$-plane for the cases $a=1$ and $a=-1$.