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\[
\fbox{\fbox{$%
\begin{array}{c}
\textrm{\textsc{Stationary points and Optimisation}} \\ 
\textrm{\textsc{MP231/209/291: Assignment 2}}
\end{array}
$}}
\]
\vspace{1cm}
\begin{enumerate}
\item Find and classify the stationary points of each of the following
functions:

\begin{enumerate}
\item $f\left( x,y\right) =x^{3}+y^{3}-\frac{3}{2}x^2-3y+3$,
\item $f\left(x,y\right) =xye^{-\left(x^{2}+y^{2}\right)}$.
\end{enumerate}
In each case calculate the value of $f(x,y)$ at the stationary points.

\item Use the \emph{Lagrange multiplier method} to find the stationary points
of each of the following functions subject to the corresponding constraints:

\begin{enumerate}
\item  $f\left(x,y,z\right) =x+z+3$\quad \textrm{subject to} \quad
$x^{2}+y^{2}+z^{2}=4$,

\item $f\left(x,y\right) =x^{2}+y^{2}+20$\quad \textrm{subject to} \quad
$x^{2}+8xy+7y^{2}=225$.
\end{enumerate}
In each case calculate the value of $f$ at the stationary points.

\item Use the Lagrange multiplier method to determine the minimum distance from the origin to any point on the surface defined by
$$
2x+4y+6z = 8.
$$

\item   Use the Lagrange multiplier method to determine the minimum distance from the point $(3,-2,1)$ to any point on the surface defined by
$$
x+2y-z = 2.
$$
\end{enumerate}

\vspace{.5cm}
\textbf{All questions are worth the same mark.} 
\vspace{.5cm}

\noindent {\bf Important note}\\
 No late solutions will be considered. The work you hand in must be your own. You may not copy any part of your assignment from another person or source. If you are collaborating on one or more questions include the name(s) of your collaborator(s).
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