Consider the function $f(x,y)=y{x}^2+4x^2+y^2-4$ over the region $D=\{(x,y)$:$|x|2,|y|2\}.$

Find the critical points $off.$

$(a)$ How many critical points does $f$ have when its domain $is$ not restricted?

$(b)$ How many $of$ the critical points are located $in$ the interior $ofDDfat$ the critical points $inDD,$ these values will $be$ the same.$f=$

Min $f=$

$(d)$ The region $D$ has four boundary lines: left, right, upper and lower.

Rewrite the formula for $f$ when restricted $to$ each $of$ these boundary lines.

Left and right: $f(y)=$

Lower: $f(x)=$

Upper: $f(x)=$

$(e)$ Find the maximum and minimum values $offon$ each boundary segment.

Max $on$ left$-$right segment: $f=$

Min $on$ left$-$right segment: $f=$

Max $on$ lower segment: $f=$

Min $on$ lower segment: $f=$

Max $on$ upper segment: $f=$

Min $on$ upper segment: $f=$

$(f)$ Find the $absolute$ maximu$\frac{m}{m}$inimum $off$ over $Dby$ comparing the interior maximu$\frac{m}{m}$inimum with the boundary maximu$\frac{m}{m}$inimum.

Maximum over $D$:$f=$

Minimum over $D$:$f=$