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Question

Consider the following in respect of the function $$f(x) = | x- 3 |$$ :
1. $$f(x)$$ is continuous at $$x = 3$$
2. $$f(x)$$ is differentiable at $$x = 0$$.
Which of the above statements is/are correct ?


A
1 only
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B
2 only
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C
Both 1 and 2
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D
Neither 1 nor 2
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Solution

The correct option is C Both 1 and 2
$$\displaystyle \lim _{ x\rightarrow { 3 }^{ - } }{ f\left( x \right)  } =\displaystyle \lim _{ x\rightarrow { 3 }^{ - } }{ -\left( x-3 \right)  } =-\left( 3-3 \right) =0$$

$$\displaystyle \lim _{ x\rightarrow { 3 }^{ + } }{ f\left( x \right)  } =\displaystyle \lim _{ x\rightarrow { 3 }^{ + } }{ \left( x-3 \right)  } =3-3=0$$
$$f\left( 3 \right) =\left| 3-3 \right| =0$$
LHL$$=$$RHL $$=f\left( x \right) \Rightarrow $$continuous at $$x=3$$
$$f'\left( x \right) =\begin{cases} -1;\quad x<3 \\ 0;\quad x=3 \\ 1;\quad x>3 \end{cases}$$
$$f'\left( x \right) =-1$$ for $$x<3$$
$$\Rightarrow f'\left( x \right) =-1$$ for $$x=0$$
$$\Rightarrow f\left( x \right) $$ is differentiable at $$x=0$$

Mathematics

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