Describe the method of Lagrange multipliers for finding extrema of a function $f(x, y, z)$ subject to the constraint that $g(x, y, z)=c$.

Illustrate the method by finding the maximum and minimum values of $x y$ for points $(x, y, z)$ lying on the ellipsoid

$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$$

with $a, b$ and $c$ all positive.