- #1

karush

Gold Member

MHB

- 3,269

- 5

$\textsf{Find all local extreme values of the given function and identify each}$

$\textsf{as a local maximum,local minimum,or saddlepoint}$

$\textit{$f$ has a} \textbf{ saddle point } \textit{at $(a,b)$ if}$

\begin{align*} \displaystyle

f_{xx}f_{yy}-fxy^2&\ge 0 \text{ at } (a,b)

\end{align*}

\begin{align*} \displaystyle

f_5(x,y)&=2xy-3x+3y\\

f_x&=2y-3\\

f_y&=2x+3\\

f_x(3/2)&=2y-3=0\\

f_y(-3/2)&=2x+3=0\\

f\left[-\frac{3}{2},\frac{3}{2} \right]

&=2\left[-\frac{3}{2} \cdot \frac{3}{2} \right]

-3\left[-\frac{3}{2} \right]

+3\left[\frac{3}{2} \right]=\frac{9}{2}\\

f_{xx}&=2\\

f_{yy}&=2\\

f_{xx}f_{yy}&=2 \cdot 2-[2 \cdot 2]^2\\

&=4-16=-12 < 0

\end{align*}

\begin{align*} \displaystyle

\textit{Book Answer: }

f\left[-\frac{3}{2},\frac{3}{2} \right]&=\color{red}{\frac{9}{2}}

\textit{ ,Saddle point}

\end{align*}first time

so suggestions?