I have just solved a problem:
Let f:[0,+∞)→R be continuous on [0,+∞) and differentiable on (0,+∞). If limx→+∞f′(x)=0, prove that limx→+∞f(x)x=0
My questions is about the inverse: if limx→+∞f(x)x=0, which hypothesis can be added (if needed) so that we can conclude limx→+∞f′(x)=0 ?
Actually, I tried to solve the following one, but I still cannot solve it:
If f:[0,+∞)→[0,+∞) such that its second derivative is continuous, f′≤0 and |f″|≤M for some M for all x≥0, then limx→+∞f′(x)=0.
From the hypothesis, f decreases and bounded below, so f has a limit when x tends to infinity. Thus limx→+∞f(x)x=0.
My questions seems to be in a wider range than the second problem above. Any help would be appreciated.
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