Math 28

Math 28, Summer 2021 Name: Quiz 4 Instructions: There are 3 problems. Read each problem carefully. Label answers appropriately. Show all your work. Submit your work on Canvas as a single PDF under Assignments, Quiz 4 Submission. [8] 1) Sketch the graph of y = f (x) satisfying the following properties: The domain of f is (−∞, −2) ∪ (−2, ∞) and f is continuous on its domain. The intercepts of f are given by f (−5) = 0, f (3) = 0, f (−1) = 0 and f (0) = −3. lim f (x) = 2 and lim f (x) = ∞ x→∞ x→−2 f ′ (x) > 0 on (−∞, −2) ∪ (0, ∞) and f ′ (x) < 0 on (−2, 0) f ′′ (x) > 0 on (−5, −2) ∪ (−2, 3) and f ′′ (x) < 0 on (−∞, −5) ∪ (3, ∞) Label the axes and indicate the scale on the axes. Label all local extrema (max/min) and inflection points (IP). 1 [6] 2) Find the points where f has an absolute maximum and and an absolute minimum for the function f (x) = x4 + 4×3 on the interval [−4, 1]. [6] 3) A box is constructed with a square base and a volume of 36 cubic feet. The material for the base costs $4.50 per square foot, the material for the top costs $3.50 per square foot, and the material for the sides costs $3.00 per square foot. Find the minimum cost of constructing the box. Justify your answer using the 2nd derivative test. 2
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