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I think many people have done this exercise in mathematical analysis. I saw it from this：Question 15

Suppose $f(x)$ is a twice-differentiable real function on (-∞,+∞), and $M_0$, $M_1$,$M_2$ are the least upper bounds of |$f(x)$|, |$f^{'}(x)$|, |$f^{''}(x)$|, respectively, on (-∞,+∞). As we can see, we can prove $$M_1^2\le2M_0M_2$$ A natural question is : Is 2 the best result for this inequality?Is there a more exact result?

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    Good question! In particular, one can ask this sub-question of your question: Does there exist a bounded, twice differentiable function $f\colon\mathbb{R}\to\mathbb{R}$ such that $M_1^2=2M_0M_2$? – 2017-02-10
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    Thanks for replying! I think the example is hard to find. – 2017-02-10
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    I agree, it's not an easy question. I did _try_ to find an example. Based on a small sample, I don't think $k = 2$ is achievable. My guess is that the minimum value of $k$ for which the inequality is always satisfied is $k=1$. – 2017-02-10
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    I gave you $+1$ (it had been $-2$). I'm not sure why you got negative votes. Perhaps the down-votes were due to the fact that you showed no work. But in this case, your question is (1) clearly not a HW problem; (2) a question showing insight and curiosity; (3) simply stated and appealing, but possibly quite hard. Thus, in my opinion, the very _asking_ of the question deserves some credit. – 2017-02-10
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    By the way, for the pdf you referenced, are those _your_ solutions? – 2017-02-10
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    I'm glad for your comment.This is my first time to ask question on math stack exchange, it seems that I do think less before asking, next time I will pay more attention to this.The solutions are not mine,I just search them on google.I will keep exploring. – 2017-02-10
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    I used to think k=1 is right, but later I find an example:sin(x)/(x^2+1).For this function k is about 1.35 – 2017-02-11
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    Interesting. When I get a chance, maybe tomorrow, I'll play with it some more. – 2017-02-11

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Now I know 2 is the best result, though the example for $M_1^2=2M_0M_2$ I haven't discovered, as it requires some knowledge I have not learnt.Click here to see Landau-Kolmogorov Constants.