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Chapter 4 20 - 282 CHAPTER 4 APPLICATIONS OF...

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Unformatted text preview: 282 CHAPTER 4 APPLICATIONS OF DIFFERENTIATION 12. (a) f(m) = 5 — 33:2 +333 => f'(av) : —6m+3m2 = 390(33 — 2). Thus. f’(a:) > 0 41> m < Dora: > 2 and f’(a:) < 0 4:) 0 < a: < 2. So f is increasing on (—00.0) and (2. 00) and f is decreasing 0n (0. 2). (b) f changes from increasing to decreasing at ac = 0 and from decreasing to increasing at so = 2. Thus. f(O) : 5 is a local maximum value and f(2) = 1 is a local minimum value. (c) f"(x) = #6 + 61: 2 6(15 —1).f"(:c) > 0 <=> an > 1 and f”(x) < 0 4:) ac < 1. Thus. f is concave upward on (1. 00) and concave downward on (#00. 1). There is an inflection point at (1, 3). 13. (a) f(m) : 1:4 — 2332 + 3 :> f/(w) = 4m3 ~ 4:1: 2 4:0(302 # 1) = 433(m + 1)(£B — 1). I Interval a: —— 1 m at A 1 f’(:B) f cc < —1 — — ~ — decreasing on (—00. —1) —1 < a: < 0 —- 7 e + increasing on (—1.0) 0 < m < 1 * decreasing on (0.1) m > 1 + + increasing on (1. oo) So f is increasing on (-1. 0) and (1. 00) and f is decreasing on (—00. *1) and (0.1). (b) f changes from increasing to decreasing at cc : 0 and from decreasing to increasing at an : 71 and a: = 1. Thus. f (0) = 3 is a local maximum value and f (:1) = 2 are local minimum values. (c) f”(:lc) : 12m2 — 4 : 12(332 — %) : 12(a:+ 1/\/§)(m ~ 1/\/§). f”(a:) > 0 4:) a: < il/x/gor an > 1/\/§ and f"(:c) < 0 <:> —1/\/?_) < a: < l/x/S. Thus. f is concave upward on (—00, —\/§/3) and (x/g/3. 00) and concave downward on (—x/g/B. VET/3). There are inflection points at (::\/§/37 232). 2 2 3 2 — 2 2 6 . . . . . 14_ (a) f(:p) : $233+ 3 :> f’(m) : Wm : EFF—3); The denominator IS pos1t1ve so the Slgn of f’(m) is determined by the Sign ofas. Thus. f'(:1:) > 0 (i) a: > 0 and f'(m) < 0 (i) m < 0. So f is increasing 0n (0, 00) and f is decreasing on (—007 0). (b) f Changes from decreasing to increasing at r = 0. Thus. f (0) = 0 is a local minimum value. (:32 + 3)2(6) — 6m . 2(m2 + 3)(2x) _ 6(m2 + 3) [ac2 + 3 — 4332] (C) JUICE) * [(552 + 3)2]2 (m2 + 3)4 _ 6(3 — 312) 7 —18(w + 1)(a: # 1) _ (:52 + 3)3 ’ (m2 + 3)3 ' f”(m) > 0 4:) #1 < w < 1 and f”(m) < 0 (i) m < -1 ora: > 1. Thus. f is concave upward on (~171) and concave downward on (—00, #1) and (1, 00). There are inflection points at (i1. i). 15. (a)f(w)::n—2sinzon(0.37r) :> f’(m)=1‘2cosa:.f'(a:)>0 41> 1—2cosm>0 (I) c0sx<§ 4:) §<w<5for7§<m<37r.f’($)<0 (i) cosm>§ 4:) O<x<§or%’<w<%‘.$ofis ~ ' 7 - ' 7r 57r 77r increasmg on (g Lg) and (——31r . 3n). and f is decreasmg on (0, g) and (—3 . —3 ). . . . . . . fl 7. _ 77, (b) f changes from 1ncreasrng to decreasmg at a: : 5?". and from decreasmg to increasmg at ac — E and at m # —3 . 1r Thus. “53") : E31 + \/§ % 6.97 is alocal maximum value and f(g) : g — \/§ % #068 and “7%) : 7?" — J57 m 5.60 are local minimum values. (c) f"(m) : 251nm > 0 4:) 0 < :c < Trand 27r < a: < 371'. f”(a:) < 0 4:) 7r < a: < 2%. Thus. f is concave upward on (0., 7r) and (271', 3n). and f is concave downward on (7r. 2n). There are inflection points at (7r7 7r) and (2W. 2W). ...
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