一道的高数题,求解

Question

设函数f(x)在x=0的某邻域二阶可导,且lim( sinx+f(x)x)/(x^3)=0
求f(0).f'(0),f"(0)

我试着把fx设成o(x^2)-1还有用taylor公式换用,但行不通,求高人指点.

Answer 1

设函数f(x)在x=0的某邻域二阶可导，则有f(x)=f(0)+xf'(0)+1/2x^2f''(y) (y介于0，x之间)
lim( sinx+f(x)x)/(x^3)
=lim( x+1/6x^3+o(x^4)+x(f(0)+xf'(0)+1/2x^2f''(y)))/(x^3)
=lim[(1+f(0))x+x^2f'(0)+x^3(1/6+1/2f''(y))+o(x^4)/(x^3)=0
若使上式成立，必须有
1+f(0)=0，f(0)=-1
f'(0)=0
1/6+1/2f''(y)=0,f''(0)=-1/3 (令y趋于0，该题应该说f具有连续的二阶导数，否则不成立)

Answer 2

f(x)x=-sinx+o(x^3)=-x+x^3/6+o(x^3)
f(x)=-1+x^2/6+o(x^2);
所以f(0)=-1;f'(0)=0;f''(0)=(1/6)*2=1/3本回答被提问者采纳