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## 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?