Question:
Let $f: R \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{c}2 \sin \left(-\frac{2 x}{2}\right), \text { if } x<-1 \\ \left|a x^{2}+x+b\right|, \text { if }-1 \leq x \leq 1 \\ \sin (\pi x) \quad \text { if } x>1\end{array}\right.$ If $\mathrm{f}(\mathrm{x})$ is continuous on $\mathrm{R}$, then a $+\mathrm{b}$ equals :
Correct Option: , 2
Solution:
If $f$ is continuous at $x=-1$, then $f\left(-1^{-}\right)=f(-1)$
$\Rightarrow 2=|a-1+b|$
$\Rightarrow|a+b-1|=2 \ldots \ldots(i)$
similarly
$f\left(1^{-}\right)=f(1)$
$\Rightarrow|a+b+1|=0$
$\Rightarrow a+b=-1$