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Assume that $f \in C^2 ([1,4])$ and for any $ \epsilon_1 , \epsilon_2 \in (0,1) $, there exsits $\lambda \in (1+ \epsilon_1 , 4 - \epsilon_2) $ such that $ | f'( \lambda ) | \leqslant | f(4 - \epsilon_2 ) | + | f ( 1 + \epsilon_1 ) | $( In fact this is by using the mean value theorem). Anyway, if we assume this statement, how can I derive the following? $$ \forall x \in [1,4], \;\;|f'(x)| \leqslant | f(4 - \epsilon_2 )| + | f( 1 + \epsilon_1 )| + \int_1^4 |f''(t)| dt $$

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We can write, by the fundamental theorem of calculus, that \begin{align} |f'(x)|&\leq |f'(x)-f'(\lambda)|+|f'(\lambda)|\\ &=\left|\int_{\lambda}^xf''(t)dt\right|+ |f'(\lambda)|\\\ &\leq \int_{\lambda}^x\left|f''(t)\right|dt+|f'(\lambda)|\\\ &\leq \int_1^4|f''(t)|dt+|f'(\lambda)|\\\ &\leq\int_1^4|f''(t)|dt+|f(4-\varepsilon_2)|+|f(1+\varepsilon_1)|. \end{align}

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Thank you very much!! –  Misaj Jun 16 '12 at 18:51

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