# Proving an Inequality about a function.

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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