Chapter 4: Partial Differentiation
Section 4.10: Optimization on Closed Domains
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Example 4.10.5
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Find the extreme values of the function on the domain consisting of the region bounded by the parabola and the line .
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Solution
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Mathematical Solution
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The unconstrained extrema are found by solving the equations . Thus, the equations
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are solved for the one real solution ≐ , which is outside .
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Along the constraint , find the extrema of by solving the equation
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for its one real solution . Hence, find ≐ .
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Along the constraint , find the extrema of by solving the equation
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for . Hence, find = ≐ 4.2219.
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To take the constraints two at a time, obtain the values of at and , the intersections of the constraints and . Hence, find = ≐ 5.35 and = ≐ 4.75.
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Figures 4.10.5(a - c) show, respectively, that portion of the surface residing over the domain , the graph of , and the graph of .
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use plots in
module()
local p1,p2,p3,f;
f:=(x^2*y-3*x+5*y^2+5)/20;
p1:=plot3d(f,x=-2..2,y=x^2..4,style=surfacecontour,lightmodel=none);
p2:=plot3d(0,x=-2..2,y=x^2..4,shading=xy);
p3:=display(p1,p2,lightmodel=none,axes=frame,tickmarks=[5,5,6],orientation=[-35,70,0],scaling=constrained);
print(p3);
end module:
end use:
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Figure 4.10.5(a) Surface on
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module()
local p1,F;
F:=(6*x^4-3*x+5)/20;
p1:=plot(F,x=-2..2,0..6,labels=[x,typeset(Eval(` f`,y=x^2))]);
print(p1);
end module:
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Figure 4.10.5(b)
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module()
local p1,G;
G:=(4*x^2-3*x+85)/20;
p1:=plot(G,x=-2..2,0..6,labels=[x,typeset(Eval(` f`,y=4))]);
print(p1);
end module:
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Figure 4.10.5(c)
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Table 4.10.5(a) lists the local and global extrema found.
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Critical point not in
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Local and global minimum
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Local minimum
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Local and global maximum
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Local maximum
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Table 4.10.5(a) Local and global extrema of on
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Maple Solution - Interactive
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Initialize
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Context Panel: Assign Function
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Obtain any unconstrained extrema
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Calculus palette: Partial-differentiation operator
Press the Enter key.
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Context Panel: Solve≻Solve (explicit)
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Extrema along the constraint
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Context Panel: Differentiate≻With Respect To≻
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Context Panel: Solve≻Solve
There is just one real solution to the equation , namely, .
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Context Panel: Evaluate and Display Inline
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Context Panel: Approximate≻5 (digits)
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=
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Extrema along the constraint
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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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Context Panel: Approximate≻5 (digits)
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=
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Take the constraints two at a time
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Obtain the values of at and , the intersections of the constraints and .
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻5 (digits)
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=
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Numeric solution via the
The Optimization Assistant is no longer listed in Tools≻Assistants. It is now found in Tools≻Tutors≻Optimization
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Type and press the Enter key.
Context Panel: Optimization≻Optimization Assistant
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Add the constraints shown in Figure 4.10.5(d).
In the Add Constraint box,
write and press Add
write and press Add
In the Add Bound box: and press Add.
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Figure 4.10.5(e) shows the Optimization Assistant finding the global minimum.
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Figure 4.10.5(f) shows the Optimization Assistant finding the global maximum.
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Figure 4.10.5(d) Dialog for adding constraints
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Figure 4.10.5(e) Finding the minimum
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Figure 4.10.5(f) Finding the maximum
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Maple Solution - Coded
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Initialize
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Obtain any unconstrained extrema
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Use the diff command to obtain the equations , and the solve command to obtain their solution. Note the use of the explicit option.
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The real solution is outside .
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Take the constraints one at a time
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Use the diff and solve commands to obtain critical numbers for evaluated along the single constraints.
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=
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Take the constraints two at a time
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Obtain the values of at and , the intersections of the constraints and .
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=
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=
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Obtain the extrema numerically with the Maximize and Minimize commands in the Optimization package. These command find local, not global, extrema. Inclusion of the initialpoint option helps in the location of global extrema. Note how the region is described in these commands.
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