$f(x)=\left\{\begin{array}{cc}2 \sin \left(-\frac{\pi x}{2}\right), & \text { if } x<-1 \\ \left|a x^{2}+x+b\right|, & \text { if }-1 \leq x \leq 1 \\ \sin (\pi x), & \text { if } x>1\end{array}\right.$
If $f(x)$ is continuous on $R,$ then $a+b$ equals ..... .
$\Rightarrow f\left(1^{-}\right)=f(1)=f\left(1^{+}\right)$
$|a+1+b|=\lim _{x \rightarrow 1} \sin (\pi x)$
$|a+1+b|=0 \Rightarrow a+b=-1 ....(1)$
$\Rightarrow$ Also $f\left(-1^{-}\right)=f(-1)=f\left(-1^{+}\right)$
$\lim _{x \rightarrow-1} 2 \sin \left(\frac{-\pi x}{2}\right)=|a-1+b|$
$|a-1+b|=2$
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 $(3) \Rightarrow a+b=-1$
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$(S_1)$ there exists $\mathrm{x}_{1}, \mathrm{x}_{2} \in(2,4), \mathrm{x}_{1}<\mathrm{x}_{2}$, such that $f^{\prime}\left(x_{1}\right)=-1$ and $f^{\prime}\left(x_{2}\right)=0$
$(S_2)$ there exists $\mathrm{x}_{3}, \mathrm{x}_{4} \in(2,4), \mathrm{x}_{3}<\mathrm{x}_{4}$, such that $f$ is decreasing in $\left(2, x_{4}\right)$, increasing in $\left(x_{4}, 4\right)$ and $2 f^{\prime}\left(x_{3}\right)=\sqrt{3} f\left(x_{4}\right)$.
Then