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Explain solution rd sharma class 12 chapter 9 Differentiability exercise Multiple choice question, question 26

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HINTS: understand the definition of continuity and differentiability.

GIVEN: f(x)=\left\{\begin{array}{cc} 1 & x \leq-1 \\ |x| & -1<x<1 \\ 0 & x \geq 1 \end{array}\right.


At x=-1

\begin{aligned} &\lim _{x \rightarrow-1^{-}} f(x)=\lim _{x \rightarrow-1} 1=1 \\ &\lim _{x \rightarrow-1^{+}} f(x)=\lim _{x \rightarrow-1}|x|=\lim _{x \rightarrow-1}-x=1 \end{aligned}

So f(x) is continuous at x=-1

Now at x=1

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


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

f(x) is not continuous at x=+1.Hence not differentiable at x=1

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