ConcepTest • Section 4.4 • Question 1 Given that f ‘(x) is continuous everywhere and changes from negative to positive at x = a, which of the following statements must be true? a is a critical point of f (x) f (a) is a local maximum f (a) is a local minimum f ‘(a) is a local maximum f ‘(a) is a local minimum ConcepTest • Section 4.4 • Question 1

ConcepTest • Section 4.4 • Answer 1 (a) and (c). a is a critical point and f (a) is a local minimum. COMMENT: Follow-up Question. What additional information would you need to determine whether f (a) is also a global minimum? Answer. You need to know the values of the function at all local minima. In addition, you need to know the values at the endpoints (if any), or what happens to f(x) as x --> ±∞. ConcepTest • Section 4.4 • Answer 1

ConcepTest • Section 4.4 • Question 2 On the same side of a straight river are two towns, and the townspeople want to build a pumping station, S. See Figure 4.7. The pumping station is to be at the river’s edge with pipes extending straight to the two towns. Which function must you minimize over the interval 0 ≤ x ≤ 4 to find the location for the pumping station that minimizes the total length of the pipe? ConcepTest • Section 4.4 • Question 2

ConcepTest • Section 4.4 • Answer 2 (b). By the Pythagorean Theorem, Distance from Town 1 to Distance from Town 2 to so the sum of these distances is given by (b). COMMENT: Follow-up Question. What function must be minimized if construction of the pipeline from Town 1 to the river is twice as expensive per foot as construction of the pipeline from Town 2 to the river and the goal is to minimize total construction cost? Answer. ConcepTest • Section 4.4 • Answer 2

ConcepTest • Section 4.4 • Question 3 Figure 4.11 shows the curves y = √x, x = 9, y = 0, and a rectangle with it sides parallel to the axes and its left end at x = a. What is the maximum perimeter of such a rectangle? 9.25 14 18 18.5 20 ConcepTest • Section 4.4 • Question 3

ConcepTest • Section 4.4 • Answer 3 (d) We wish to choose a to maximize the perimeter of the rectangle with corners at (a,√a) and (9,√a). The perimeter of this rectangle is given by the formula We are restricted to 0 ≤ a ≤ 9. To maximize this perimeter, we set dP/da = 0, and the resulting perimeter is greater than the perimeter if a = 0 or a = 9. ConcepTest • Section 4.4 • Answer 3

ConcepTest • Section 4.4 • Answer 3 ANSWER (cont’d) We have P = 18 if a = 0 and P = 6 if a = 9. All that remains is to find where dP/da = 0: We have P = 18.5 when a = 1/4 . The maximum perimeter is 18.5 when the dimensions of the rectangle are 0.5 by 18.75 ConcepTest • Section 4.4 • Answer 3

ConcepTest • Section 4.4 • Question 4 You wish to maximize the volume of an open-topped rectangular box with a square base x cm by x cm and surface area 900 cm2. Over what domain can the variable x vary? 0 < x < 900 0 < x < 30 0 < x < √450 0 < x < ∞ ConcepTest • Section 4.4 • Question 4

ConcepTest • Section 4.4 • Answer 4 (b) We have x > 0 because x is a length. The square base has area x2 which must be less then 900, the surface area of the entire box. Hence 0 < x < √900 = 30. COMMENT: When working an optimization word problem, students are sometimes so focused on finding a formula for the function to be optimized that they can forget to identify the domain over which the optimization takes place. ConcepTest • Section 4.4 • Answer 4