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Suppose that $f\in C^2[0,1]$ and with bounds $|f(x)|\leq a,|f(x)''|\leq b,\forall x\in [0,1]$.Do we have any estimate on $|f(x)'|$,and how to get it?I heard a result saying that if for some $x_0\in [0,1],|f(x_0)'|\leq d$,then $|f(x)'|\leq 2\sqrt{ab}+d$.Is it right?How to prove or disprove it?I would appreciate it if someone would give me some hints on this problem.

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The Mean Value Theorem says that $|f'(x)-f'(y)|\le b|x-y|\le b$ for all $x,y\in[0,1]$. Thus, if $|f'(x_0)|>2a+b$ for some $x_0\in[0,1]$, then $|f'(y)|>2a$ for all $y\in[0,1]$. The Mean Value Theorem also says that there is a $y_0\in(0,1)$ so that $|f'(y_0)|=|f(1)-f(0)|\le 2a$.

Thus, we must have that $|f'(x)|\le 2a+b$ for all $x\in[0,1]$.

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