Question

Let $d$ be the number of integers in the range of the function $f(x)=\begin{cases}4, & \text { if }-4 \leq x< -2 \\ |x|, & \text { if }-2 \leq x< 7 \\ \sqrt{x}, & \text { if } 7 \leq x< 14\end{cases}.$.
Also roots of $P ( x )= x ^2+ mx -4 m +20$ are $\alpha$ and $\beta$. If $\alpha<\frac{ d -3}{4}< \frac{ d -3}{2}< \beta$ and the smallest integral value of $m$ is $k$, then find the value of $( k -5)$.

Answer 1

Range of $f(x)$ is $[0,7)$
Hence, $d =7$
Now, one root of $P ( x )$ is less than 1 and other root greater than 2 .
Hence, $P (1)<0 \Rightarrow 21-3 m <0 \Rightarrow m >7$
and $P (2)<0 \Rightarrow 24-2 m <0 \Rightarrow m >12$
Hence, $m >12$.
$\therefore$ Least integral value of $m$ is 13
$\Rightarrow( k -5)=8$. Ans. $]$