MATH1013 Calculus I, 2013-14 Spring Tutorial Worksheet 8

MATH1013 Calculus I, 2013-14 Spring
Tutorial Worksheet 8: Extrema
Name:
(T1A)
ID No.:
Tutorial Section:
Complete at least THREE questions from the following questions.
(Solution of this worksheet will be available at the course website the week after.)
1. (Demonstration) (page 238, Q. 21) Sketch the graph of a continuous function f on [0, 4] satisfying
f 0 (1) and f 0 (3) are undefined; f 0 (2) = 0; f has a local maximum at x = 1; f has local minimum at
x = 2; f has an absolute maximum at x = 3; and f has an absolute minimum at x = 4.
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2. (Demonstration) (p. 238, Q. 46) Find the critical points of f (x) = x 2 − x2 on [− 2, 2]. Determine
the absolute extreme of f when they exist. Give a quick sketch of the f to verify your findings.
3. (Demonstration) (p.238, Q. 53) Suppose a tour guide has a bus that holds a maximum of 100 people.
Assume his profit for taking n (could be non-integer) people on a city tour is P (n) = n(50 − 0.5n) − 100.
(a) How many people should the guide take on a tour to maximize the profit?
(b) Suppose the bus holds a maximum of 45 people. How many people should be taken on a tour to
maximize the profit?
4. (Demonstration) (p.251, Q. 11) Sketch a continuous function f on (−∞, ∞) that possesses the properties: f 0 (x) < 0 on (−∞, 2); f 0 (x) > 0 on (2, 5); f 0 (x) < 0 on (5, ∞).
5. (Demonstration) (p.252, Q. 57) Determine the intervals of concave up/down, and to identify the
inflection point(s) of f (x) = x4 − 2x3 + 1.
6. (Class work) (page 238, Q. 22) Sketch the graph of a continuous function f on [0, 4] satisfying f 0 (x) = 0
at x = 1 and x = 3; f 0 (2) is undefined; f has an absolute maximum at x = 2; f has neither a local
maximum nor a local minimum at x = 1; and f has an absolute minimum at x = 3.
Answer
7. (Class work) (p. 238, Q. 47) Find the critical points of f (x) = 2x3 − 15x2 + 24x on [0, 5]. Determine
the absolute extreme of f when they exist. Give a quick sketch of the f to verify your findings.
Answer
1
8. (Class work) (p. 238, Q. 48) Find the critical points of f (x) = x arcsin x on [−1, 1]. Determine the
absolute extreme of f when they exist. Give a quick sketch of the f to verify your findings.
Answer
9. (Class work) (p.238, Q. 52) A sales analyst determines that the revenue from sales of fruit smoothies
is given by R(x) = −60x2 + 300x where x is dollars charged per item, for 0 ≤ x ≤ 5.
(a) Find the critical point of the revenue function?
(b) Determine the absolute maximum value of the revenue function and the price that maximizes the
revenue?
Answer
10. (Class work) (p.251, Q. 12) Sketch a continuous function f on (−∞, ∞) that possesses the properties:
f 0 (−1) is undefined; f 0 (x) > 0 on (−∞, −1); f 0 (x) < 0 on (−1, ∞).
Answer
11. (Class work) (p.252, Q. 58) Determine the intervals of concave up/down, and to identify the inflection
point(s) of f (x) = −x4 − 2x3 + 12x2 .
Answer
2