Determine whether the function $\displaystyle \mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-6 \mathrm{x}+3$ is increasing or decreasing in [$\displaystyle 4,6$].

$$\begin{aligned} & \mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-6 \mathrm{x}+3, \mathrm{x} \in[4,6] & \text { We have } \mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{x}-6 & \text { For all } \mathrm{x} \text { such that } 4<\mathrm{x}<6,2<2 \mathrm{x}-6<6 & \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})>0 \text { for all } \mathrm{x} \in(4,6) \end{aligned}Hence, f is increasing over [$\displaystyle 4,6$].