Re: [分析] 求範圍
※ 引述《jazzter (阿里巴巴你媽媽)》之銘言：
: 想請教
: [2(a1^2+a2^2+a3^2+a4^2) / (a1+a2+a3+a4)^2] -
: [(a1^3+a2^3+a3^3+a4^3) / (a1+a2+a3+a4)^3]
we may assume w + x + y + z = 1
A = 2 (w^2+x^2+y^2+z^2) - (w^3+x^3+y^3+z^3)
Let w+x=r and r is constant
K = 2w^2+2x^2-w^3-x^3
= 2w^2+2(r-w)^2-w^3-(r-w)^3
= 2r^2 - r^3 - (4-3r)w(r-w)
w(r-w) = -(w- r/2)^2 + r^2/4 <= r^2/4
Hence, min K occurs at w = x = r/2
max K occurs at w=0,r, that is, max K = 2r^2 - r^3
Similarly,
y+z=1-r
H = 2y^2+2z^2-y^3-z^3
min H occurs at y = z = (1-r)/2
max H occurs at w = 0,(1-r), that is, max H = 2(1-r)^2 - (1-r)^3
min K + min H =(1/4)(5r^2-5r+3)
Hence min A = 7/16 occurs at w=x=y=z=1/4
max K + max H = r^2 - r + 1
Hence, max A = 1 occurs at w=1, x=y=z=0, or other permutations
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07/18 17:55, , 1F
07/18 17:55, 1F
※ 編輯: JohnMash 來自: 27.147.57.77 (07/19 02:07)
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