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  • please solve rd sharma class 12 chapter 9 Differentiability exercise Multiple choice question, question 6 maths textbook solution

please solve rd sharma class 12 chapter 9 Differentiability exercise Multiple choice question, question 6 maths textbook solution

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(a)

HINTS: Understand the  definition of continuity and differentiability

GIVEN: f(x)=e^{-\left | x \right |}

SOLUTION:

f(x)= \begin{cases}e^{-x} & x>0 \\ e^{x} & x<0\end{cases}

At x=0

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

Hence, f(x) is continuous everywhere

LHD at x=0

\begin{aligned} \lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0} &=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{0-h-0} \\ &=\lim _{h \rightarrow 0} \frac{e^{-h}-1}{-h} \\ &=\lim _{h \rightarrow 0} \frac{e^{-h}}{-h} \quad\left[\frac{0}{0} \text { form }\right] \\ &=-e^{-0} \\ &=-1 \end{aligned}

RHD  at x=0

\begin{aligned} \lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0} &=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{0+h-0} \\ &=\lim _{h \rightarrow 0} \frac{e^{-h}-1}{h} \\ &=\lim _{h \rightarrow 0} \frac{-e^{-h}}{1} \quad\left[\frac{0}{0} \text { form }\right] \\ &=-e^{-0} \\ &=-1 \end{aligned}

LHD \neq RHD

Hence, f(x) is  differentiable at x=0

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