\documentclass[a4paper,12pt]{article}
\usepackage{epsfig}
\begin{document}
\parindent=0pt

\begin{center}
\textbf{Applications of Partial Differentiation}

\textit{\textbf{Extremes}}
\end{center}

\textbf{Question}

Find and classify the critical points of the function

$$f(x,y,z)=xy+x^2z-x^2-y-z^2$$


\textbf{Answer}

\begin{eqnarray*}
f(x,y,z) & = & xy+x^2z-x^2-y-z^2\\
f_1(x,y,z) & = & y+2x(z-1)\\
f_2(x,y,z) & = & x-1\\
f_3(x,y,z) & = & x^2-2z
\end{eqnarray*}
The only critical point is $(1,1,\frac{1}{2})$. We have
\begin{eqnarray*}
D & = & f(1+h,1+k,1/2 +m)-f(1,1,1/2)\\
& = & 1+h + k+hk+\frac{1+2h+h^2}{2}+(1+2h+h^2)m\\
& & -1-2h-h^2-1-k-\frac{1}{4}-m-m^2- \left ( -\frac{3}{4} \right )\\
& = & \frac{h^2(2m-1)+2h(k+2m)-2m^2}{2}
\end{eqnarray*}

If $m=h$ and $k=0$, then $\displaystyle D = \frac{h^2(1+2h)}{2}>0$ for
small $|h|$.

If $h=k=0$, then $D=-m^2<0$ for $m\ne0$.

Thus $f$ has a saddle point at $(1,1,1/2)$.


\end{document}