※ 引述《ntnusliver (炸蝦大叔~~)》之銘言：
: p p
: 1. For 1≦p＜∞ , show that the L norm arises from a scalar product of L (R)
: if and only if p=2.
: 2 3 4
: 2. Let p(x)=a0+ a1x + a2 x +a3 x + a4 x be a polynomail with real
: coefficients such that p'(x)>0 p"(x)>0 , p'''(x)>0 on [0,1].
: 3
: If t=(t1,t2,t3) in R , t1+t2+t3=1 0≦ti≦1
: Minimize g(t):= p(t1)+p(t2)+p(t3).
自問自答一下
由於 p"(x)>0 => p(x) is a convex function on [0,1]
apply Jensen's Inequality
p(1/3)= p( (1/3)t1 + (1/3)t2 + (1/3)t3 ) ≦ (1/3)p(t1)+(1/3)p(t2)+(1/3)p(t3)
=> 3 p(1/3) ≦ p(t1)+p(t2)+p(t3) = g(t)
最小值是 3 p(1/3)
( 似乎沒用到p'''(x)>0 )
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※ 編輯: ntnusliver 來自: 122.120.42.182 (05/23 11:46)
※ 編輯: ntnusliver 來自: 122.120.42.182 (05/23 11:53)