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\begin{document}

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\bf{Question}

\quad  For each of the following functions $f:\br^2\to\br$ find
the critical points and decide if they are nondegenerate or
degenerate. Classify the nondegenerate critical points as maxima,
minima or saddles. Sketch contours for $f$, taking care to include
those which pass through the critical points.

\medskip

\begin{tabular}{lllll}$f(x,y)=$\hspace{1cm}&(i)&$xy(x^4+y^4+1)$&(iv)&$y^2-3yx^2+2x^4$\\
&(ii)& $x^3+y^2-3x$ &(v)& $x^3-3xy^2$\\ &(iii)& $\sin x+\sin
y+\cos(x+y)$\hspace{1cm} &(vi)& $x^4+y^4-2(x^2+y^2)$
\end{tabular}



\bf{Answer}

\begin{description}
\item{(i)}
$xy(x^4+y^4_1)$. Vanishes on axes $x=0$, $y=0$ only.

\begin{center}
\epsfig{file=342-3A-1.eps, width=45mm}
\end{center}

Saddle point at $(0,0)$ (Hessian matrix $\left ( \begin{array}{cc}
0 & 1 \\ 1 & 0 \end{array} \right )$).

$f(x,y)=xy\times(\textrm{positive quantity which increases as }\|
(x,y) \| \to\infty)$.

\item{(ii)}
$(x^3-3x)+y^2$:

\begin{center}
\epsfig{file=342-3A-2.eps, width=45mm} \ \ \
\epsfig{file=342-3A-3.eps, width=45mm}

\epsfig{file=342-3A-4.eps, width=50mm}
\end{center}

Hence minimum at $(1,0)$, saddle at $(-1,0)$.

Dots where $f>2$, white where $f<2$.

\item{(iii)}
$\displaystyle \left. \begin{array}{rl} \frac{\partial f}{\partial
x} & = \cos x - \sin (x+y)\\ \frac{\partial f}{\partial y} & =
\cos y - \sin (x+y)
\end{array} \right \}$ vanish when $\begin{array}{rl}
\cos x & = \cos y\\ x & =  2n\pi \pm y
\end{array}$

Then

$x=y$ gives $x=y=\frac{\pi}{6}$, $\frac{5\pi}{6}$,
$\frac{3\pi}{2}$; and $\frac{\pi}{2}$mod$2\pi$.

$x=-y$ gives $x=-y=\frac{\pi}{2}$, $\frac{3\pi}{2}$mod$2\pi$.

\begin{center}
\epsfig{file=342-3A-5.eps, width=60mm}

o=max, $\bullet$=min, $\times$=saddle

($f(x,y)$ symmetric about the line $x=y$.)
\end{center}

Note also
\begin{eqnarray*}
x = \frac{\pi}{2} & \Rightarrow & f=1\\ y = \frac{\pi}{2} &
\Rightarrow & f=1\\ \textrm{and }x+y=2\pi &\Rightarrow & f=1
\end{eqnarray*}

Hessian matrices as follows

$\begin{array}{rlrl} \ds \left ( \frac{\pi}{6}, \frac{\pi}{6}
\right ) : & \ds \left ( \begin{array}{cc} -1 & -\frac{1}{2} \\
-\frac{1}{2} & 1
\end{array} \right ) \textrm{ max} &

\ds \left ( \frac{5\pi}{6}, \frac{5\pi}{6} \right ) : & \ds \left
( \begin{array}{cc} -1 & -\frac{1}{2} \\ -\frac{1}{2} & -1
\end{array} \right ) \textrm{ max}\\

\ds \left ( \frac{3\pi}{2}, \frac{3\pi}{2} \right ) : & \ds \left
( \begin{array}{cc} 2 & 1 \\ 1 & 2
\end{array} \right ) \textrm{ min} &

\ds \left ( \frac{\pi}{2}, \frac{\pi}{2} \right ) : & \ds \left (
\begin{array}{cc} 0 & 1 \\ 1 & 0
\end{array} \right ) \textrm{ saddle}\\

\ds \left ( \frac{\pi}{2}, \frac{3\pi}{2} \right ) : & \ds \left (
\begin{array}{cc} -2 & -1 \\ -1 & 0
\end{array} \right ) \textrm{ saddle } &

\ds \left ( \frac{3\pi}{2}, \frac{\pi}{2} \right ) : & \ds \left (
\begin{array}{cc} 0 & -1 \\ -1 & -2
\end{array} \right ) \textrm{ max} \end{array}$

Dotted region is where $f>1$, (max=$\frac{3}{2}$)

White region is where $f<1$, (min=$-\frac{3}{2}$)

Note that $f(x,0)=\sqrt{2}\sin (x + \frac{\pi}{4})$.

\item{(iv)}

\begin{eqnarray*}
y^2-3yx^2+2x^4 & = & \left ( y - \frac{3}{2} x^2 \right )^2\\ & =
& (y-x)(y-2x^2) \\ & = & 0\\ \textrm{when } \frac{1}{2}x^2 & = &
\pm \left ( y = \frac{3}{2} x^2 \right )\\ \Rightarrow y & = & x^2
\ \textrm{or } 2x^2
\end{eqnarray*}

\begin{center}
\epsfig{file=342-3A-6.eps, width=50mm}
\end{center}

The only critical point is $(0,0)$.

Dotted region: $f>0$

White region: $f<0$.

\item{(v)}
$x^3-3xy^2 = x(x-\sqrt{3}y)(x+\sqrt{3}y)$

\begin{center}
\epsfig{file=342-3A-7.eps, width=50mm}
\end{center}

Dots: $>0$, White: $<0$.

\item{(vi)}
9 critical points, where $x,y=0,1,-1$.

\begin{center}
\epsfig{file=342-3A-8.eps, width=40mm} \ \ \
\epsfig{file=342-3A-9.eps, width=40mm}
\end{center}

Just add to obtain

\begin{center}
\epsfig{file=342-3A-10.eps, width=70mm}

o=max, $\bullet$=min ($\times 4$), $\times$=saddle ($\times 4$)

Symmetry about $y=x$ and $y=-x$
\end{center}
\end{description}

All the critical points are non-degenerate, except in cases (iv),
(v). In both these cases the origin is the only critical point,
and is degenerate.


\end{document}