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
1 Answers
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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 $x\in[0,1]$.