Question

Let $f:R\to R$ be defined as
$f(x)=\{\begin{array}{ll}2sin\left(-\frac{\pi x}{2}\right)& \text{if }x<1\\ |a{x}^2+x+b|& \text{if }-1\le x\le 1\\ sin(\pi x)& \text{if }x>1\end{array}$
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 $R$
$\Rightarrow f\left(1^{-}\right)=f(1)=f\left(1^{+}\right)$
$|a+1+b|=\underset{x\to 1}{\lim }\sin (\pi x)$
$|a+1+b|=0\Rightarrow a+b=-1\dots (1)$
$\Rightarrow$ Also $f\left(-1^{-}\right)=f(-1)=f\left(-1^{+}\right)$
$\underset{x\to -1}{\lim }2\sin \left(\frac{-\pi x}{2}\right)=|a-1+b|$
$|a-1+b|=2$
Either $a-1+b=2$ or $a-1+b=-2$
$a+b=3\dots (2)$ or $a+b=-1\dots$ (3)
from (1) and (2) $\Rightarrow a+b=3=-1$( reject )
from (1) and (3) $\Rightarrow a+b=-1$