Question

Number of integral value(s) of $k$ for which the equation $4x^2-16x+k=0$ has one root lie between $1$ and $2$ and other root lies between $2$ and $3$, is

Answer 1

$f\left(x\right)=4x^2-16x+k=0$

One root between 1 & 2

other root between 2 & 3.

According to rolle's theorem.

$f\left(1\right).b\left(2\right)<0$

$(-12+K)(-16+K)<0$

$(k-12)(K-16)<0$

$Kϵ(12,16)$

Also, $f\left(2\right)b\left(3\right)<0$

$-16+K(-12+K)<0$

$(K-16)(K-12)<0$

$Kϵ(12,16)$

Intersection both is $Kϵ(12,16)$

Total integral values possible of $K=3\{13,14,15\}$