Prove f(0)=f'(0)=f"(0)=0 in (-1,1) interval w/ Positive M

[Janez25]

Homework Statement

Suppose f:(-1,1)→R is three times differentiable on the interval. Assume there is a positive value M so that ⎮f(x)⎮ ≤ M⎮x⎮³ for all x in (-1,1). Prove that f(0)=f'(0)=f"(0)=0.

Homework Equations

The Attempt at a Solution

My professor started us off,
⎮f(0)⎮≤M(0)=0; f(0)=0
f'(0)=lim as x→0 [(f(x)-f(0))/x-0 = lim as x→0 [f(x)/x].
I know that ⎮f(x)/x⎮≤ 1/⎮x⎮(M⎮x⎮³
≤ Mx²
Which means that f'(0) = 0
I also know that the next step is to find f"(0) = lim as x→0 [(f'(x)-f'(0))/x-0].
I need to know if f'(x) = 1/x?

 

[snipez90]

f'(x) is not 1/x. Think about what f(x) is for x > 0 and for x < 0. Then think about what f'(x) should be for x > 0 and for x < 0. This should help you continue the problem. When you have to show f''(0) = 0, again think about what f''(x) should be for x > 0 and for x < 0.