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Chapter 4 163 - 10 11 12 CHAPTER 4 PROBLEMS PLUS 425 After...

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Unformatted text preview: 10. 11. 12. CHAPTER 4 PROBLEMS PLUS 425 After expanding and canceling terms, we get flat) 2 are”? — x1173 — 1333112 + mlxz — $2332 + (mg) 2 flatflm * x) + $301: — $1) + 372(x1 — m2)] f’(:v) f’(:z:) = 0 => 2$(w1 — x2) : as? — mg => asp = %($1 +$2)- l 7m? + x3 + 2m(m1 — m2)]. f”(at) = %[2(w1 — 1:2)l : m1 — 372 < 0 since 232 > 131‘ 2 ll WP) = arc? l%(w2 — $01)] +m§ [an — 331)] + flan +£2)2($1 — 352)) = % B062 - $1)(fvi + $3) — flan ~ mm + 952?] : £062 — 031M200? + m3) — (as? + 2mm + 3%)] : a“ ”W? -2m2 +26%) = m w mm m)? : gm mm my Zita 79603 To put this in terms ofm and b, we solve the system 3; = 17% and y : man + 1). giving us an? — mml — b : O :> 2:1 : %(m — x/m2—+4b). Similarly, m2 2 %(m + VHF—+411). The area is then an — m1)3 2 am)?’ and is attained at the point P(mp,$§j) : P(%m, inf). Note: Another way to get an expression for f (ac) is to use the formula for an area of a triangle in terms of the coordinates ofthe vertices: f(m) : %[(m2m§ * 331333) + (3mm2 v 323:?) + ($533 — 502562)]. If f”(a:) > 0 for all :17, then f’ is increasing on (—00, 00), so f’(0) must be greater than f’(71). But f’(0) = 0 < % = f'(—1). so such a function cannot exist. f(.1’) : (a2 + a i 6) c0822: + (a — 2):r —l— cosl => f’($) : — (a2 +a — 6) sin2m (2) + (a — 2). The derivative exists for all :3, so the only possible critical points will occur where f’(m) : 0 4:) 2(a — 2)(a + 3) sin 2:1: : a — 2 (1) either a = 2 or 2(a + 3) sin 23: : 1. with the latter implying that 1 sin 23: : m Since the range of sin 2:5 is [—1, 1]. this equation has no solution whenever either a. 1 1 . A . t 7 5 m < —1 or m > 1. Solv1ng thCSC mequalltles. We get *5 < (l < *5. To sketch the region {($31) | 2mg 3 Ian — yl g $2 + 3/2}, we consider two cases. Case I: a: 2 y This is the case in which (:0, y) lies on or below the line y : m. The double inequality becomes 2mg 3 x ~ 3/ g m2 + 312. The right—hand inequality holds if and only if $2 — a; —|— y2 + y 2 O {i} (m — %)2 + (y + %)2 Z % <:> (w, y) lies on or outside the Circle with radius % centered at G, —%). ...
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