→ Annihilator :令f(x)=x^2-xsinx/π也可... 10/21 14:39
推 laba5566 :太感謝了 A大 <(_ _)> 10/22 01:45
※ 引述《laba5566 (最愛56家族 啾咪)》之銘言：
: f : R_+ -> R_+ f(0)=0
: suppose f(x)/x is strictly increasing function for all x > 0
: prove or disprove f(x) is convex function
: 我覺得應該是正確的
: 不過嘗試許久證不出來 >"<
: 能請板上高手賜教嗎
: 感謝
Counter-example:
2 xsin(x)
f(x) = x - ---------
2
1
f'(x) = 2x - -(sinx-xcosx)
2
1 xsin(x)
f"(x) = 2 - -(cosx-cosx+xsinx) = 2 - ---------
2 2
Note f"(2π+π/2) = 2 - 5π/4 < 0.
But f(x)/x is strictly increasing
d f(x) cos(x)
since --[----] = 1 - -------- > 0.5
dx x 2
--
～by Jackary P.～
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