BACKGROUND MATERIAL

For a function $g: \Re^n \rightarrow \Re$, the gradient is denoted

$$\nabla g := \left( \begin{array}{c} \frac{\partial g}{\part... ... \cdots\\ \frac{\partial g}{\partial x_n} \end{array} \right).$$

Suppose $f:\Re \rightarrow \Re$ is infinitely differentiable at $x=a.$ The Taylor Series of $f$ about $a$ is:

$$f(a) + f^\prime(a) (x-a) + \frac1{2!}f^{\prime \prime}(a)(x-a)^2 + \frac1{3!}f^{\prime \prime \prime}(a)(x-a)^3 + \cdots.$$