Consider the following function on the set D$.$

f$($x$,$ y$)=$ x$2+$ y$2+$ x$2$y $+6,$ D $=$

$($x$,$ y$)$

$|$x$|1,|$y$|1$

Find the following derivatives.

fx$($x$,$ y$)$

$=$

$2$x$+2$xy

fy$($x$,$ y$)$

$=$

$2$y$+$x$2$

Find the critical point.

$($x$,$ y$)=$

$0,0$

Find the value of f at this critical point.

Find the minimum and maximum of f on each segment of the boundary.

minimummaximumy $=1$ x $=1$ y $=1$ x $=1$

Find the absolute maximum and minimum values of f on the set D$.$

D $=$

$($x$,$ y$)$

$|$x$|1,|$y$|1$

absolute maximum value absolute minimum value