Chapter 4 26 - 288 CHAPTER 4 APPLICATlONS 0F...

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Unformatted text preview: 288 CHAPTER 4 APPLICATlONS 0F DIFFERENTIATlON (b) Local minimum values f(:l:\/§) = —9. (d) local maximum value f(0) = O (c)f”(:c)=12a:2 — 12 : 12(gc2 — 1) > 0 c» :32 >1 4:) [ml >1 4:) 32> lora: < —1.sofisCUOn(—oo,—1).(1,oo) and CD on (*1.1). Inflection points at (:1, —5) 36. (a) g(x) : 200 + 8x3 + 9:4 :> g’(:n) = 24952 + 42:3 = 4m2(6 + m) : 0 when a; : —6 and when m = 0. g'(a:) > 0 <=> a: > —6 (m 31$ 0) and g'(:c) < 0 (i) at < —6. so 9 is decreasing On (—00, -6) and g is increasing on (—6, 00). with a horizontal tangent at an : 0. (b) g(-6) : —232 is a local minimum value. (d) There is no local maximum value. (c) g"(m) : 483: + 12:52 : 1250(4 + at) : 0 when a: : —4 and when m:0.g"(a:)>0 4:) $<—4ora:>0andg”(:1c)<0 4:) ~4 < a: < 0. so 9 is CU on (—00. —4) and (0. 00). and g is CD on (A4, 0). lnflection points at (—4, —56) and (0. 200) 37. (a) h(m) = 3x5 1 5m3 + 3 => h’(a;) : 1524 1 15:52 : 157:2(x2 1 1) = 0 when m = 0, :1. Since 15m2 is nonnegative. h'(ac) > 0 4:, 3:2 > 1 <=> |x| > 1 4:) :v > 1 orm < —1. so h is increasing on (—00. —1) and (1. 00) and decreasing on (#1., 1). with a horizontal tangent at ac : 0. (b) Local maximum value h(—1) : 5, local minimum value h(1) : 1 (c) h"(m) : 603:3 1 303: : 3012(22132 — 1) (d) : 60m<m+ ii) (a: — %) => h"(:v)>0whena:> fior—fi <x<OtsohisCUon 1 (efifi) and ($.00) and CD on (#00. —%) and (0., :5). lnflection points at (07 3) and (ii , 3 :1: gx/i) [about (70.71. 4.24) and (0.71, 1.76)]. ...
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