Chapter 3: Applications of Differentiation
Section 3.8: Optimization
Example 3.8.2
Show that among all rectangles with a fixed area, the square has the minimum perimeter.
Solution
Analysis
Figure 3.8.2(a) shows a labeled rectangle whose perimeter is 2 w+2 h, and whose area w h is fixed at A.
The constraint equation is gw,h≡w h−A=0.
Solve the constraint equation for, say, h=A/w and write the perimeter of the rectangle as 2w+A/w.
Maximize the objective function Fx=2 x+A/x.
p1:=plottools[rectangle]([1,4],[5,1],style=line): p2:=plots:-textplot({[3,.7,typeset(w)],[.7,2.5,typeset(h)]},font=[Lucinda,18]): plots:-display(p1,p2,scaling=constrained, axes=none);
Figure 3.8.2(a) Labeled diagram of a rectangle
Computation
Define the objective function Fx
Control-drag Fx=…
Context Panel: Assign Function
Fx=2 x+A/x→assign as functionF
Find critical numbers
Write the equation for critical numbers. Press the Enter key.
Context Panel: Solve≻Obtain Solutions for≻x
F′x=0
2−2Ax2=0
→solutions for x
A,−A
Second-Derivative test
Write F″A Context Panel: Evaluate and Display Inline
F″A = 4A
Of the two critical numbers found, namely, w=±A, only the positive root is meaningful since w is a dimension. The purist would claim that in addition to the constraint equation g=0, there are additional constraints, namely, that the variables must be nonnegative.
Since F″A>0, the perimeter is a minimum at w=A=h, this second equality being determined from the constraint w h=A=hA.
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