Example 4.10.1

Find the extreme values of the function $f\left(x\,y\right)\=x y\+1\/2$ on the domain $R$ consisting of the interior and boundary of the triangle whose sides are the $x$- and $y$-axes, and the line $x\+y\=1$.

Example 4.10.2

Find the extreme values of the function $f\left(x\,y\right)\=2-x^2-y^2\+2x\+2y$ on the domain $R$ consisting of the interior and boundary of the triangle whose sides are the $x$- and $y$-axes, and the line $x\+y\=9$.

Example 4.10.3

Find the extreme values of the function $f\left(x\,y\right)\=x^2-xy\+y^2\+1$ on the domain $R$ consisting of the triangular region whose edges are the $y$-axis, and the lines $y\=x$ and $y\=4$.

Example 4.10.4

Find the extreme values of the function $f\left(x\,y\right)\=\left(x^2-xy\+y^2\+4y\+10\right)\/10$ on the domain $R$ consisting of the interior and boundary of the disk whose radius is 4, and whose center is at the origin.

Example 4.10.5

Find the extreme values of the function $f\left(x\,y\right)\=\left(5 \+x^2 y-3 x\+5 y^2\right)\/20$ on the domain $R$ consisting of the region bounded by the parabola $y\=x^2$ and the line $y\=4$.

Example 4.10.6

Find the extreme values of the function $f\left(x\,y\right)\=3x^2\+3xy\+2y^2\+5x-4 y$ on the domain $R$ consisting of the interior and boundary of the ellipse whose equation is $2x^2-4xy\+3y^2\+4x-5y-1\=0$.