4.6 Discussion of Taylor’s theorem

Consequently, when $(a,b)\,$ is not a stationary point, the sum of $f(a,b)\,$ and the term in braces gives a good approximation to $f(a+h,b+k)\,$ near to $(a,b)\,$. However, when $(a,b)\,$ is a stationary point, the term in braces is zero and $f(a+h,b+k)\,$ may be approximated near to $(a,b)\,$ by a quadratic expression in $h\,$ and $k\,$ with coefficients given by the second-order partial derivatives of $f\,$.

This observation will be useful in determining whether stationary points are local maxima, local minima, or another type of stationary point called a saddle point.

Before proving it, we make the following observation, using the Chain rule: if $g(x,y)$ is a function and $G(t)=g(a+th,b+tk)$ then

Concisely, $G^{\prime}(t)=hg_{x}+kg_{y}$.