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Chapter 4 80

# Chapter 4 80 - 342 El CHAPTEB4 APPLICATIONS OF...

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Unformatted text preview: 342 El CHAPTEB4 APPLICATIONS OF DIFFERENTIATION 12. From the graph. it appears that f increases on (~52, —1.0) and (1.0, 5.2) and decreases on (—27r, —5.2), (—1.0, 1.0), and (5.2. 27r): that f has local maximum values off(71.0) % 0.7 and f(5.2) % 7.0 and local minimum values of f(—5.2) z —7.0 and f(1.0) z —0.7‘. that f is CU on (—27r. —3.1) and (0. 3.1) and CD on (—31.0) and (3.1. 27r). and that f has IP at (0, 0). (~31, —3.1) and (3.1, 3.1). ﬂan) : a: — 2sinx => f/(x) : 1 — 2cos m, which is positive (f is increasing) when cosac < % that is, on (—5?7r —%) and (% i371), and negative (f is decreasing) on (~27r7 —5?7'). (—g g) and (5?” . 271'). By the FDT, f has local maximum values off(—§) : g + \/3andf(5?") = 5?" + \/3. and local minimum values off(—5?") : is?" — x/3and f(§) : 7% * x/3. f’(m) : 1 * 2coszv => f”(:1:) 2 28mm. which is positive (f is CU) on (—271'7 ~7r) and (07 7r) and negative (f is CD) on (in. 0) and (7r, 271'). f has IP at (0, 0), (771', in) and (7r, 7r). 13. (a) f(a:) : x2 In x. The domain off is (0.00). l -0.25 0 1.75 IE7- —0.25 . Ina: H . 1/m . m2 . ' 2 _ _l _1 —— =0.Th relsaholeat 0,0. 0” .1339 1” .1313. W2 .331. -W .334 2 e I > (c) It appears that there is an IP at about (0.2, —0.06) and a local minimum at (0.6, —0.18). f(ac) = \$2 lna: i 1/2 f’(m) : 902(1/zc) + (lncc)(2ac) : 93(2lnm + 1) > 0 <2> lnx > —% 4:) a: > e’ , so f is increasing on (1/\/E, oo), decreasing on (0,1/\/E). By the FDT. f(1/\/E) : —1/(2e) is a local minimum value. This point is approximately (0.6065, 70.1839), which agrees with our estimate. f”(ac)=\$(2/m)+(2lnm+1)=2lnm+3>0 41> lnat>—% 41> x>e’3/2.sofisCUon (6—3/2,oo) and CD on (me-3”). IP is ((3/2, —3/(2e3)) x (0.2231, 70.0747). 4" 14. (a) ﬂat) : {tel/I. The domain off is (700,0) U (0. oo). W anal/.2) ' 1(3”: ' e 21' ——=lime1/I:oo.soa::0isaVA. (b) \$213+ \$6 (3212+ 1/(13 xiii): —1/\$2 x—>0+ Also lim weal/’3 : 0 since l/a: —> ~00 => (Bl/I —> 0. \$40— ...
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