Exercise 3.3

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MA111: Prepared by Dr.Archara Pacheenburawana 31 Exercise 3.5 1. Use the given graph of f to find the intervals of concavity. (a) y 20 10 −3 −1 1 3 x (b) y 1 −4 −2 2 4 x (c) y 10 5 −2 2 4 x −5 −10 (d) y 10 5 −3 −1 1 2 3 x −5 2. Given the following functions. • Find the interval of increase and decrease. • Find the local maximum and minimum values. • Find the intervals of concavity and the inflection points.

• • • • MA111: Prepared by Dr.Archara Pacheenburawana 32 • Use the above information to sketch the graph. (a) f(x) = 2x 3 −3x 2 −12x (b) f(x) = x 4 −6x 2 (c) f(x) = 3x 5 −5x 3 +3 (d) f(x) = x √ x 2 +1 (e) f(x) = x 1/3 (x+3) 2/3 (f) f(x) = sin 2 x Answer to Exercise 3.5 1. (a) CU on (−∞,−1)∪(1,∞); CD on (−1,1) (b) CU on (−∞,0); CD on (0,∞) (c) CU on (1,∞); CD on (−∞,1) (d) CU on (−1,0)∪(1,∞); CD on (−∞,−1)∪(0,1) 2. (a) Inc. on (−∞,−1)∪(2,∞); dec. on (−1,2); loc. max. f(−1) = 7; loc. min. f(2) = −20; CU on ( 1,∞); CD on (−∞, 1); IP 2 2 (1 2 ,−13) 2 y −10 5 1 x −20 (b) Inc. on (− √ 3,0)∪( √ 3,∞); dec. on (−∞,− √ 3)∪(0, √ 3); loc. min. f(± √ 3) = −9; loc. max. f(0) = 0; CU on (−∞,−1)∪(1,∞); CD on (−1,1); IP (±1,−5) y −1 1 • • x • • −10 (c) Inc. on (−∞,−1)∪(1,∞); dec. on (−1,−1); loc. max. f(−1) = 5; loc. min. f(1) = 1; CU on (−1/ √ 2,0)∪(1/ √ 2,∞); CD on (−∞,−1/ √ 2)∪(1/ √ 2,∞); IP (0,3), (±1/ √ 2,3∓ 8√ 7 2)

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