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# ISM_T11_C04_D - 224(c Chapter 4 Applications of Derivatives...

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224 Chapter 4 Applications of Derivatives (c) 35. (a) h(x) 2x 4x h (x) x 4x 4 (x 2) a critical point at x 2 h [ and œ Ê œ œ Ê œ Ê œ ± ! # x 3 # w # # w h(0) 0 no local maximum, a local minimum is 0 at x 0 œ Ê œ (b) no absolute maximum; absolute minimum is 0 at x 0 œ (c) 36. (a) k(x) x 3x 3x 1 k (x) 3x 6x 3 3(x 1) a critical point at x 1 œ Ê œ œ Ê œ \$ # w # # k ] and k( 1) 0, k(0) 1 a local maximum is 1 at x 0, no local minimum Ê œ ± œ œ Ê œ " ! w (b) absolute maximum is 1 at x 0; no absolute minimum œ (c) 37. (a) f(x) 2 sin f (x) cos , f (x) 0 cos a critical point at x œ Ê œ œ Ê œ Ê œ x x x x 2 3 # # # # # # w w " " ˆ ‰ ˆ ‰ ˆ ‰ 1 f [ ] and f(0) 0, f 3, f(2 ) local maxima are 0 at x 0 and Ê œ ± œ œ œ Ê œ ! # # Î\$ w 1 1 1 1 1 ˆ È 2 3 3 1 1 at x 2 , a local minimum is 3 at x œ œ 1 1 1 3 3 2 È (b) The graph of f rises when f 0, falls when f 0, w w and has a local minimum value at the point where f w changes from negative to positive.

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Section 4.3 Monotonic Functions and the First Derivative Test 225 38. (a) f(x) 2 cos x cos x f (x) 2 sin x 2 cos x sin x 2(sin x)(1 cos x) critical points at œ Ê œ œ Ê # w x , 0, f [ ] and f( ) 1, f(0) 3, f( ) 1 a local maximum is 1 at œ Ê œ ± œ œ œ Ê ! 1 1 1 1 1 1 w x , a local minimum is 3 at x 0 œ „ œ 1 (b) The graph of f rises when f 0, falls when f 0, w w and has local extreme values where f 0. The w œ function f has a local minimum value at x 0, where œ the values of f change from negative to positive. w 39. (a) f(x) csc x 2 cot x f (x) 2(csc x)( csc x)(cot x) 2 csc x 2 csc x (cot x 1) a critical œ Ê œ œ Ê # w # # a b a b point at x f ( ) and f 0 no local maximum, a local minimum is 0 at x œ Ê œ ± œ Ê œ ! Î% 1 1 1 4 4 4 w 1 1 ˆ ‰ (b) The graph of f rises when f 0, falls when f 0, w w and has a local minimum value at the point where f 0 and the values of f change from negative to w w œ positive. The graph of f steepens as f (x) . w Ä „ _ 40. (a) f(x) sec x 2 tan x f (x) 2(sec x)(sec x)(tan x) 2 sec x 2 sec x (tan x 1) a critical point œ Ê œ œ Ê # w # # a b at x f ( ) and f 0 no local maximum, a local minimum is 0 at x œ Ê œ ± œ Ê œ Î# Î# Î% 1 1 1 4 4 4 w 1 1 1 ˆ ‰ (b) The graph of f rises when f 0, falls when f 0, w w and has a local minimum value where f 0 and the w œ values of f change from negative to positive. w 41. h( ) 3 cos h ( ) sin h [ ] , ( ) and ( 3) a local maximum is 3 at 0, ) ) 1 ) 1 œ Ê œ Ê œ !ß \$ # ß Ê œ ! # ˆ ‰ ˆ ‰ ) ) # # # w w 3 a local minimum is 3 at 2 œ ) 1 42. h( ) 5 sin h ( ) cos h [ ] , ( 0) and ( 5) a local maximum is 5 at , a local ) ) 1 ) 1 1 œ Ê œ Ê œ ß Ê œ !
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ISM_T11_C04_D - 224(c Chapter 4 Applications of Derivatives...

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