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please solve rd sharma class 12 chapter 9 Differentiability exercise Multiple choice question, question 2 maths textbook solution

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HINTS: Understand the  definition of continuity and differentiability

GIVEN: f(x)=\sin ^{-1}(\cos x)


Check the continuity at x=0


\begin{aligned} \lim _{x \rightarrow 0^{-}} f(x) &=\lim _{h \rightarrow 0} f(0-h) \\ &=\lim _{h \rightarrow 0} \sin ^{-1}(\cos (0-h)\\ &=\lim _{h \rightarrow 0} \sin ^{-1}(\cos h) \\ \end{aligned}

                     \begin{aligned} &=\sin ^{-1}(\cos 0) \\ &=\sin ^{-1}1 \\ &=\frac{\pi}{2} \\ \end{aligned}

\begin{aligned} \lim _{x \rightarrow 0^{+}} f(x) &=\lim _{h \rightarrow 0} f(0+h) \\ &=\lim _{h \rightarrow 0} \sin ^{-1}(\cos h) \\ \end{aligned}

                     \begin{aligned} &=\sin ^{-1}(\cos 0) \\ &=\sin ^{-1} 1 \\ &=\frac{\pi}{2} \end{aligned}

As \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)

therefore f(x) is continuous at x=0

check the differentiability at x=0

LHD  at x=0

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