## Can a Lipschitz continuous function become convex after adding a large enough quadractic function? [closed]

Let $f : [a,b] \to \mathbb{R}$ be Lipschitz continuous. Is $f(x) + cx^2$ convex for some $c$ large enough?

Of course, this is true when $f$ is in $C^2$, but how about only Lipschitz now.

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This question is probably not appropriate for mathoverflow, but I can answer if you ask on math.stackexchange.com. – Noah Stein Feb 9 2011 at 17:21
This looks like a homework problem. (I am tempted to give you a hint but I think that would not be the best thing to do.) Even if this is not a homework problem, mathoverflow is ment for research level questions - you might want to go to math.stackexchange.com – Tapio Rajala Feb 9 2011 at 17:21
Also when you ask it there, you may want to change "Can" in the title to "Must" or "Will" to align with the body of the question. – Noah Stein Feb 9 2011 at 17:23
The answer in "NO", take $$f(x)=x{\cdot}\sin(1/x)$$ and define $f(0)=0$. – Anton Petrunin Feb 9 2011 at 18:01
Anton, doesn't your example have unbounded derivative? So how can it be Lipschitz? – Joel David Hamkins Feb 9 2011 at 18:36
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