If both the roots of the quadratic equation  $x^2-mx+4=0$ are real and distinct and they lie in the interval $[1,5],$  then m lies in the interval

$$\begin{gathered} x^2-mx+4=0 \alpha ,\beta \in [1,5] \left(1\right)\text{D}>0\Rightarrow m^2-16>0 \Rightarrow m\in (-\infty ,-4)\cup (4,\infty ) \\ \left(2\right)f\left(1\right)\ge 0\Rightarrow 5-m\ge 0\Rightarrow m\in (-\infty ,5] \left(3\right)f\left(5\right)\ge 0\Rightarrow 29-5m\ge 0\Rightarrow m\in (-\infty ,\frac{29}{5}] \end{gathered}$$

$$\begin{gathered} \left(4\right)1<\frac{-b}{2a}<5\Rightarrow 1<\frac{m}{2}<5\Rightarrow m\in (2,10) \\ \Rightarrow m\in (4,5] \end{gathered}$$