$f(x)$ is continuous
$ \Rightarrow \mathop {\lim }\limits_{x \to {1^ - }} f\left( x \right) = \mathop {\lim }\limits_{x \to {1^ + }} a + {\cos ^{ - 1}}\left( {x + b} \right) = f\left( x \right)$
$ \Rightarrow - 1 = a + {\cos ^{ - 1}}\left( {1 + b} \right)$
${\cos ^{ - 1}}\left( {1 + b} \right) = - 1 - a\,\,\,\,\,\,...\left( 1 \right)$
$f(x)$ is differentiate
$ \Rightarrow LHD = RHD$
$ \Rightarrow - 1 = \frac{{ - 1}}{{\sqrt {1 - {{\left( {1 + b} \right)}^2}} }}$
$ \Rightarrow 1 - {\left( {1 + b} \right)^2} = 1$
$ \Rightarrow b = - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,....\left( 2 \right)$
From $\left( 1 \right) \Rightarrow {\cos ^{ - 1}}\left( 0 \right) = - 1 - a$
$\therefore - 1 - a = \frac{\pi }{2}$
$a = - 1 - \frac{\pi }{2}$
$a = \frac{{ - \pi - 2}}{2}\,\,\,\,\,\,...\left( 3 \right)$
$\therefore \frac{a}{b} = \frac{{\pi + 2}}{2}$
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Statement$-1$ If $f R \rightarrow R$ and $c \in R$ is such that $f$ is increasing in $(c - \delta , c)$ and $f$ is decreasing in $(c, c + \delta )$ then $f$ has a local maximum at $c$. Where $\delta$ is a sufficiently small positive quantity.
Statement $-2$ Let $f (a, b) \rightarrow \,R, c \in (a, b)$. Then $f$ can not have both a local maximum and a point of inflection at $x = c.$
Statement $-3 $ The function $f (x) = x^2 | x |$ is twice differentiable at $x = 0.$
Statement $-4$ Let $f [c - 1, c + 1] \rightarrow [a, b]$ be bijective map such that $f$ is differentiable at $c$ then $f^{-1}$ is also differentiable at $f (c)$.