If both the roots of the quadratic equation x ^ 2 - mx + 4 = 0 are...

Given roots of ${x}^{2} - mx + 4 = 0$ are distinct and lies in $\left(1, 5\right),$ Following conditions must be true

$( i ) D > 0 ⇒ {m}^{2} - 16 > 0$

$⇒ \left(m - 4\right)\left(m + 4\right) > 0$

$⇒ m \in \left(- \infty, - 4\right) \cup \left(4, \infty\right)$

$( ii ) f\left(1\right) > 0 ⇒ m < 5$

$( iii ) f\left(5\right) > 0 ⇒ m < \dfrac{29}{5}$

$( iv ) 1 < - \dfrac{b}{2a} < 5 ⇒ 2 < m < 10$

Taking intersection of all conditions, we get $m \in \left(4, 5\right)$