Question

Let $f:\left[0,\frac{5}{2}\right)\to \left(\frac{1}{2},3\right]$ be a function defined as $f(x)=\frac{[x]+1}{\{x\}+1}$ where $[]$ and $\{\}$ represent the greatest integer function and fractional part. Given that $\min \cdot \left(\underset{x\to 1^{-}}{\lim }f(x),\underset{x\to 1^{+}}{\lim }f(x)\right)=\frac{m}{p}$, G.C.D. $(m,p)=1$ and maximum of the values of $x$, at which $f$ is discontinuous, is equal to $f(k).$ Find the value of $p^2-k^2{m}^2.$

Answer 1

Answer: 3

$f$ is discontinuous at $x=1,2$.
$\Rightarrow \max$. (the values of $x$ at which $f$ is discontinuous)
$=\max .(1,2)=2$
$=f(1)$
$\Rightarrow k=1$
$\min \cdot \left(\underset{x\to 1^{-}}{\lim }f(x),\underset{x\to 1^{+}}{\lim }f(x)\right)=\min \cdot \left(\frac{1}{2},2\right)$
$=\frac{1}{2}$
$\Rightarrow m=1,p=2$
$\Rightarrow p^2-k^2{m}^2=4-(1{)}^2(1{)}^2=3$