※ 引述《diow1 (小玉)》之銘言：
: 已知 0 ≦ x+y ≦ 2π 求函數 f(x,y)= sinx + siny - sin(x+y) 的最大及小值 ???
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max = 3√3 / 2, min = -3√3 / 2
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x+y=2u, x-y=2v, x=u+v, y=u-v,
0 ≦ u ≦ π,
K= sin x + sin y - sin(x+y)
=sin(u+v) + sin(u-v) -sin(2u)
= 2 sin u cos v - 2 sin u cos u
= 2 sin u (cos v - cos u)
sin u >= 0
when u fixed, max K occurs at max cos v = 1
min K occurs at min cos v = -1
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max K = 2 sin u (1- cos u)
Let cos u = t
(max K)^2/4 = sin^2 u (1-cos u)^2=(1-t^2)(1-t)^2=(1+t)(1-t)^3
(3+3t)(1-t)^3<={[(3+3t+3(1-t)]/4}^4=(6/4)^4=81/16
max K = 3√3 / 2 when 1-t = 3+3t, t=-1/2, cos u = -1/2, sin u = √3 /2
2u=x+y=4π/3, 2v=x-y=0, x=y=2π/3
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min K = 2 sin u (-1-cos u )=-2 sin u (1+cos u) = -M
min K occurs at max M
Let cos u = t
(max M)^2/4 = sin^2 u (1+cos u)^2=(1-t^2)(1+t)^2=(1-t)(1+t)^3
(3-3t)(1+t)^3<={[(3-3t+3(1+t)]/4}^4=(6/4)^4=81/16
max M = 3√3 / 2 when 1+t = 3-3t, t=1/2, cos u = 1/2, sin u = √3 /2
min K = -3√3 / 2
2u=x+y=2π/3, 2v=x-y=2π, x=4π/3, y=-2π/3
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◆ From: 112.104.143.69
※ 編輯: JohnMash 來自: 112.104.143.69 (07/08 11:54)