作者LuisSantos (^______^)
看板trans_math
标题Re: [考古] 政大应数4
时间Mon Jan 29 02:10:16 2007
※ 引述《spss520 (剩下一个月啦)》之铭言：
: Does there exist a real-valued function f(x) defined on (0,∞)
: ,
: satisfying f(1)=1 and xf (x^2)=1 for all x＞0
: 谢谢大家
Yes
(x)(f'(x^2)) = 1 , f(1) = 1 , x > 0
1
f'(x^2) = ---
x
Let x^2 = t Then x = √t (x > 0)
1
f'(t) = ---- , f(1) = 1
√t
1
f(t) = ∫----- dt
√t
= (2)(√t) + c
f(1) = (2)(√1) + c = 1
2 + c = 1 => c = 1 - 2 = -1
f(t) = (2)(√t) - 1
=> f(x) = (2)(√x) - 1
--
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