Question

大一高数问题如图5题问题如图，求解。

Answer 1

解:由(x-1)f(x+1/x-1)+f(x)=x (1)
令y=x+1/x-1, 于是x=y+1/y-1,带入到(1)式得
(2/(y-1))*f(y) + f(y+1/y-1)=y+1/y-1 (2)
将变量y换成x得
(2/(x-1))*f(x) + f(x+1/x-1)=x+1/x-1 (3)
等式两边乘以 x-1,得
2f(x)+(x-1)*f(x+1/x-1)=x+1 (4)
联立(1)，(4)解得
f(x)=1(x不能等于1)
3. f(x)=根号下(x-1)
因为 x^2-x+1>0 恒成立，
所以x^2>x-1
|x|>根号下(x-1)
或 根号下(x-1)<|x|
取k=1,即得
|f(x)|<|x|
4. f(x)=x/(x^2+x+1)
|f(x)|=|x/(x^2+x+1)|≤k|x|
等价于:|1/(x^2+x+1)|≤k
等价于:x^2+x+1≥1/k
x^2+x+1=x^2+x+1/4+3/4=(x+1/2)^2+3/4≥3/4(=1/k)
取k=4/3
则 f(x)≤4/3|x|