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Chapter 4: Partial Differentiation
Section 4.10: Optimization on Closed Domains
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Example 4.10.3
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Find the extreme values of the function on the domain consisting of the triangular region whose edges are the -axis, and the lines and .
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Solution
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Mathematical Solution
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Figure 4.10.3(a) shows that part of the surface defined by for which the support is the domain .
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The unconstrained extrema, obtained by solving the equations , is .
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Taking the constraints one at a time requires solving three constrained optimization problems, with each side of the triangle being the constraint in each of the three problems.
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Figure 4.10.3(b), a graph of , suggests the maximum , and the minimum .
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Figure 4.10.3(c), a graph of , suggests the maximum and the minimum .
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Figure 4.10.3(d), a graph of , suggests the maximum and the minima .
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Taking the constraints two at a time requires finding the values of at the intersection of the edges of the triangle, that is, at the vertices. These values are , , so the absolute minimum is , and the absolute maximum is 17.
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>
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use plots in
module()
local f,p1,p2,p3;
f:=x^2+y^2-x*y+1;
p1:=plot3d(f,x=0..4,y=x..4, style=patchcontour):
p2:=plot3d(0,x=0..4,y=x..4,color=red):
p3:=display([p1,p2],axes=frame,orientation=[-120,75,0],tickmarks=[5,5,4],labels=[x,y,z],lightmodel=none);
print(p3);
end module:
end use:
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Figure 4.10.3(a) The surface over the triangle
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Figure 4.10.3(b) Along
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Figure 4.10.3(c) Along
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Figure 4.10.3(d) Along
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Each of the three constrained optimization problems obtained when applying the constraints one at a time is a candidate for the Lagrange multiplier method. However, in this example, the simplicity of the three optimization problems that result does not warrant the more elaborate machinery of the Lagrange multiplier approach.
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Maple Solution - Interactive
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Initialize
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Context Panel: Assign Function
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Solve the unconstrained problem
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Calculus palette: Partial-derivative operator
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Context Panel: Solve≻Solve
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Context Panel: Evaluate and Display Inline
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Apply the constraints one at a time
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Along (the -axis):
Context Panel: Differentiate≻With Respect To≻
Context Panel: Solve≻Solve
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Along :
Context Panel: Differentiate≻With Respect To≻
Context Panel: Solve≻Solve
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Along : Press the Enter key.
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Context Panel: Differentiate≻With Respect To≻
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Context Panel: Solve≻Solve
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Context Panel: Evaluate and Display Inline
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Take the constraints two at a time, that is, evaluate at the vertices of the triangle
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Maple Solution - Coded
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Initialize
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Define the objective function .
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Solve the unconstrained optimization problem
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Apply the diff command to form the equations , and the solve command to obtain their solution.
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Evaluate at the critical point.
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Apply the constraints one at a time
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Along (the -axis):
Apply the solve and diff commands.
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Along :
Apply the solve and diff commands.
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Along :
Apply the diff and solve commands to obtain the critical number along this edge.
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Apply the constraints two at a time
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<< Previous Example Section 4.10
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