Re: [分析] (sin A)^(sin A) < (cos A)^(cos A) f …

[snaredrum (好聽木琴)]

※ 引述《snaredrum (好聽木琴)》之銘言：

(sin A)^(sin A) < (cos A)^(cos A)   for 0<A<pi/4

how to prove it?

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推 icenuclear:x1*logx1 < x2*logx2 iff 0<x1<x2
→ snaredrum:x*ln x 在(0,pi/4) 不是遞增吧... if x < 1/e  f'<0
推 math1209:我沒有什麼好辦法，我用 Lagrange Multiplier Method.
按照大師的推文去做，

Let f(x,y)= x lnx - y lny      constraint x^2+y^2=1  so (2x,2y)

\nobla f = (lnx +1 , lny +1)

Hessian f 對角線是 (1/x, 1/y)  0<x,y<1

所以函數在 x=y = sqr{2}/2 has min = 0

done.

am I right?

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※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 69.143.196.235
推 math1209:yes~