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**tttcomrader** Claim: $\displaystyle f(x) = \frac {4}{3}x^{ \frac {3}{2} } $ satisfies the hypothesises but the inequality $\displaystyle |f(x)| \leq cx^2 $ do not hold in any neighborhood of the origin.

Proof so far.

Now, $\displaystyle f'(x)=2x^{ \frac {1}{2} } $, so $\displaystyle f(0)=f'(0)=0$

I need to show that in any neighborhood of the origin, $\displaystyle |f(x) > cx^2 \ \ \ \ \ \forall c>0$

Given $\displaystyle c>0$ and $\displaystyle |x| \leq a $ for some $\displaystyle a \in \mathbb {R} $, we have $\displaystyle |f(x)|=|\frac {4}{3} x^ { \frac {3}{2} } | = \frac {4}{3} |x| ^ { \frac {3}{2} } \geq \frac {4}{3} (-a)^{ \frac {3}{2} } $

But how would I get a positive number on the right hand side? Thanks.