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Show that g is bounded

Source: IMC 1997 day 2 problem 1

October 19, 2005
functioncalculusderivativereal analysisreal analysis unsolved

Problem Statement

Let f∈C3(R)f\in C^3(\mathbb{R}) nonnegative function with f(0)=f′(0)=0,f′′(0)>0f(0)=f'(0)=0, f''(0)>0. Define g(x)g(x) as follows: {g(x)=(f(x)f′(x))′forx≠0g(x)=0forx=0 \{ \begin{array}{ccc}g(x)= (\frac{\sqrt{f(x)}}{f'(x)})' &\text{for}& x\not=0 \\ g(x)=0 &\text{for}& x=0\end{array} (a) Show that gg is bounded in some neighbourhood of 00. (b) Is the above true for f∈C2(R)f\in C^2(\mathbb{R})?
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