Question

Let $f: R \rightarrow R$ be defined as
$f(x) = \begin{cases} 2 sin \left(- \frac{\pi x}{2}\right) & \text{if } x < 1 \\ |ax^{2} + x + b| & \text{if } -1 \le x \le 1 \\ sin (\pi x) & \text{if } x > 1 \end{cases} $
If $f(x)$ is continuous on $R,$ then $a+b$ equals:

• A. -3
• B. -1
• C. 3
• D. 1

Answer 1

Answer: (B)

$f( x )$ is continuous on $R$
$\Rightarrow f\left(1^{-}\right)=f(1)=f\left(1^{+}\right)$
$|a+1+b|=\displaystyle\lim _{x \rightarrow 1} \sin (\pi x)$
$|a+1+b|=0 \Rightarrow a+b=-1 \ldots(1)$
$\Rightarrow $ Also $f\left(-1^{-}\right)=f(-1)=f\left(-1^{+}\right)$
$\displaystyle\lim _{x \rightarrow-1} 2 \sin \left(\frac{-\pi x }{2}\right)=| a -1+ b |$
$|a-1+b|=2$
Either $a-1+b=2$ or $a-1+b=-2$
$a + b =3 \ldots(2)$ or $a + b =-1 \ldots$ (3)
from (1) and (2) $\Rightarrow a+b=3=-1$( reject )
from (1) and (3) $ \Rightarrow a+b=-1$