※ 引述《znmkhxrw (QQ)》之銘言：
: 第二題：
: f€C^2(a,∞) , a€R
: Let M_0,M_1,M_2 be the supremum of │f(x)│,│f'(x)│,│f''(x)│ on (a,∞)
: prove that (M_1)^2 <= 4(M_0)*(M_2)
: <Hint> Use Taylor formula：for all h>0 , x€(0,∞)
: we have f(x+h)=f(x)+f'(x)h+(1/2)(f''(c)h^2) , c€(x,x+h)
: 我湊很久了都湊不出來 試過算幾不等式 但是不等式方向就相反了
: 帶入各種的h 也都不如預期
: 謝謝~
Rudin的Principles of Mathematical Analysis 第三版中的第五章習題15有這題
不過書中給的hint更多:
If h>0, Taylor's theorem shows that
f'(x)=(1/2h)[f(x+2h)-f(x)]-hf''(ξ) for some ξ∈(x, x+2h).
因此，利用三角不等式及M_0和M_2的定義，你會得到
|f'(x)|≦hM_2+M_0/h
這樣離目標應該很接近了!!
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