Question

设f(x)/x=1的x趋于0的极限，且f''(x)>0，证明f(x)>=x

Answer 1

简单计算一下即可，答案如图所示

Answer 2

lim(x->0)f(x)/x
=1
所以
f(0)=
lim(x->0)f(x)=0
f'(0)=lim(x->0)[f(x)-f(0)]/(x-0)=
lim(x->0)f(x)/x
=1
由拉格朗日余项的泰勒公式得：f(x)=f(0)+f'(0)x+f''(ξ)/2
*x²
=x+f''(ξ)/2
*x²
所以
f(x)-x=f''(ξ)/2
*x²≥0