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.
an estimate on the derivative of a function
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calculus