Question

Let $d$ be the number of integers in the range of the function $f(x)=\{\begin{array}{ll}4,& \text{ if }-4\le x<-2\\ |x|,& \text{ if }-2\le x<7\\ \sqrt{x},& \text{ if }7\le x<14\end{array}.$.
Also roots of $P(x)=x^2+mx-4m+20$ 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

Answer: 8

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, $P(1)<0\Rightarrow 21-3m<0\Rightarrow m>7$
and $P(2)<0\Rightarrow 24-2m<0\Rightarrow m>12$
Hence, $m>12$.
$\therefore$ Least integral value of $m$ is 13
$\Rightarrow (k-5)=8$. Ans. $]$