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Question

The function $$f(x)={ e }^{ -\left| x \right|  }$$ is


A
continuous everywhere but not differentiable at x=0
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B
continuous and differentiable everywhere
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C
not continuous at x=0
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D
None of the above
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Solution

The correct option is A continuous everywhere but not differentiable at $$x=0$$
Given $$f(x)=\begin{cases} { e }^{ -x },x\ge 0 \\ { e }^{ x },\quad x<0 \end{cases}$$
$$LHL=\lim _{ x\rightarrow { 0 }^{ - } }{ f(x) } =\lim _{ x\rightarrow 0 }{ { e }^{ x } } =1$$
$$RHL=\lim _{ x\rightarrow { 0 }^{ + } }{ f(x) } =\lim _{ x\rightarrow 0 }{ { e }^{ -x } } =1$$
Also, $$f(0)={e}^{0}=1$$
$$\because$$ LHL=RHL=f(0)
$$\therefore$$ It is continuous for every value of $$x$$.
Now $$LHL$$ at $$x=0$$
$${ \left( \cfrac { d }{ dz } { e }^{ x } \right)  }_{ x=0 }={ \left[ { e }^{ x } \right]  }_{ x=0 }={ e }^{ 0 }=1\quad $$ 
$$RHD$$ at $$x=0$$
$${ \left( \cfrac { d }{ dz } { e }^{ -x } \right)  }_{ x=0 }={ \left[ { -e }^{ x } \right]  }_{ x=0 }=-1$$
So, $$f(x)$$ is not differentiable at $$x=0$$
Hence, $$f(x)={ e }^{ -\left| x \right|  }$$ is continuous every but not differentiable at $$x=0$$

Mathematics

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