$$\eqalign{
& y = {x^3} - m{x^2} - \left( {m - 6} \right)x + 3\,\,DB/\left( {0;4} \right) \cr
& y' = 3{x^2} - 2mx - m + 6 > 0\,\,\forall x \in \left( {0;4} \right) \cr
& \Leftrightarrow 3{x^2} + 6 > m\left( {2x + 1} \right)\,\,\forall x \in \left( {0;4} \right) \cr
& \Leftrightarrow g\left( x \right) = {{3{x^2} + 6} \over {2x + 1}} > m\,\,\forall x \in \left( {0;4} \right) \cr
& \Rightarrow m \le \mathop {\min }\limits_{\left[ {0;4} \right]} g\left( x \right) \cr
& g'\left( x \right) = {{6x\left( {2x + 1} \right) - 2\left( {3{x^2} + 6} \right)} \over {{{\left( {2x + 1} \right)}^2}}} = {{6{x^2} + 6x - 12} \over {{{\left( {2x + 1} \right)}^2}}} = 0 \cr
& \Leftrightarrow \left[ \matrix{
x = 1 \hfill \cr
x = - 2 \hfill \cr} \right. \cr
& g\left( 0 \right) = 6;\,\,g\left( 4 \right) = 6;\,\,g\left( 1 \right) = 3 \cr
& \Rightarrow \mathop {\min }\limits_{\left[ {0;4} \right]} g\left( x \right) = 3 \Rightarrow m \le 3 \cr} $$
