Let f

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 :

  1. (1) 3

  2. (2) - 1

  3. (3) -3

  4. (4) 1


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

$f\left(1^{-}\right)=f(1)$

$\Rightarrow|a+b+1|=0$

$\Rightarrow a+b=-1$

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