(9/30/08)Math 10A. Lecture Examples.Sections 4.3 and 4.5. Optimization and modeling†Example 1 Find the maximum and minimum values of h(x) =xx2+ 4for x ≥ 0,if they exist.Answer: [Maximum] =14• [Minimum] = 0 • (The graph is in Figure A1.)x2 4 6yy = h(x)14Figure A1Example 2 (a) What is the global maximum of y = ln x −12x2+3?(b) Does the function have a g lobal minimum?Answer: [Global maximum] = 2.5 • (The graph is in Figure A2.) (b) There is no global minimum.x1 2y2−2y = ln x −12x2+ 3Figre A2†Lecture notes to accompany S ections 4.3 and 4.5 of Calculus by Hughes-Hallett et al.1Math 10A. Lecture Examples. (9/30/08) Sections 4.3 and 4.5, p. 2Example 3 (a) Find the point on the curve y =√x2+ 1 in Figure 1 that is closest tothe point (2,0) . (Minimize the square of the distance.) (b) How far is (2,0)from the curve?x2y24y =√x2+ 1(2, 0)FIGURE 1Answer: (a) Figure A3a • The closest point on the curve is (1,√2) (Figure A3b)(b) Its dist ance t o (2, 0) is√5.x2y24y =√x2+ 1x(x,√x2+ 1)x21y24y =√x2+ 1(1,√2)Figure A3a Figure A3bExample 4 Imagine you want to make a rectangular garden using a wall as one sideand 40 feet of fence for the three other sides. What dimensions would givethe garden the maximum area?Answer: Figure A4a • The area is a maximum for w = 10 and L = 20. • (These results are corroborated bythe graph of the area as a fun ction o f th e width in Figure A4b.)Wallw wLw10 20A100200A = w(40 − 2w)Figure A4a Figure A4bSections 4.3 and 4.5, p. 3 Math 10A. Lecture Examples. (9/30/08)Example 5 A rectangular box with a square base and no to p is to be constructed sothat it has a volume of 4 cubic mete rs. How wide and how tall should thebox be to minimize the total area of the base and four sides?Answer: Figure A5a • The box should be 2 meters wide and 1 meter high • (These results are corroborated bythe graph of the area as a fun ction o f th e width in Figure A5b.)hwww2 4A10203040A = w2+16wFigure A5a Figure A5bExample 6 A store will sell100025 + x2T-shirts if they charge x dollars per shirt(1 ≤ x ≤ 20). What price would maximize the revenue f rom the shirts?Answer: The revenue is maximized if the price is $5 per shirt.Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 4.5: Examples 1 and 2‡The chapter and section numbers on Shenk’s web site refer to h is calculus manuscript and no t to the chapters and sectionsof the textbook for the