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Chapter 4: Partial Differentiation
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Section 4.10: Optimization on Closed Domains
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Essentials


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To find the extrema of a function $f$ defined over a closed domain $R$, first perform an unconstrained search. Any extrema found outside $R$ are rejected. Then, search along the boundary of $R$.

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In two variables, the boundary of $R$ could be described implicitly by the equation of a circle or ellipse. It could be described explicitly as a series of curves or lines, such as the edges of a triangle. Where boundary segments join, the nodes have to be tested for extreme values.

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In three variables, the boundary of $R$ could be described implicitly by the equation of a sphere or ellipsoid. It could be described explicitly by a set of intersecting surfaces, in which case, the curves of intersection, and the points of intersection of these curves, would have to be searched.

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In either event, when optimizing an objective function subject to a constraining curve or surface, the Lagrange multiplier technique could be used. For many of the textbook exercises met at this level, simple substitution often suffices to change the search to an unconstrained one that does not need the machinery of the Lagrange multiplier method.

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Examples


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Example 4.10.1

Find the extreme values of the function $f\left(x\,y\right)\=xyplus;1sol;2$ 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 $f\left(x\,y\right)\=\left(5plus;{x}^{2}y3xplus;5{y}^{2}\right)sol;20$ 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 $f\left(x\,y\right)\=3{x}^{2}\+3xy\+2{y}^{2}\+5x4y$ on the domain $R$ consisting of the interior and boundary of the ellipse whose equation is $2{x}^{2}4xy\+3{y}^{2}\+4x5y1\=0$.



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