Question

If the function $f\left(x\right)=2p[2x+5]+q[3x-7]$ is continuous at $x=1$, then
$($where [.] denotes greatest integer function and $p,qϵR)$

• A. $2p+q=0$
• B. $p+2q=0$
• C. $p+q=0$
• D. $10p-7q=0$

Answer 1

The correct option is A $2p+q=0$

$f\left(x\right)=(10p-7q)+2p\left[2x\right]+q\left[3x\right]$
$\underset{x\to 1^{+}}{lim}f\left(x\right)=(10p-7q)+4p+3q=14p-4q....\left[1\right]$
$\underset{x\to 1^{-}}{lim}f\left(x\right)=(10p-7q)+2p+2q=12p-5q....\left[2\right]$
Since, f(x) is continuous at x = 1.

$\therefore \underset{x\to 1^{+}}{lim}f\left(x\right)=\underset{x\to 1^{-}}{lim}f\left(x\right)$
$\Rightarrow 14p-4q=12p-5q$
$\Rightarrow 2p+q=0$