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If       

  • Option 1)

    continuous for all x, but not differentiable at x=0

  • Option 2)

    neither differentiable not continuous at x=0

  • Option 3)

    discontinuous everywhere

  • Option 4)

    continuous as well as differentiable for all x

 

Answers (1)

best_answer

As we learnt in 

Condition for differentiable -

A function  f(x) is said to be differentiable at  x=x_{\circ }  if   Rf'(x_{\circ })\:\:and\:\:Lf'(x_{\circ })   both exist and are equal otherwise non differentiable

-


 f(x)= \left\{\begin{matrix} xe^{-\left ( \frac{1}{\left | x \right |}+\frac{1}{x} \right )} \: \: \: \: \: x\neq 0\\ 0 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: x=0 \end{matrix}\right.

Since, \left | x \right |\is\ not\ differential\ at\ x=0

            \left | x \right | \: is \: continuous \: at \: x=0

\therefore \: option \: 1 \: is \: correct


Option 1)

continuous for all x, but not differentiable at x=0

This option is correct

Option 2)

neither differentiable not continuous at x=0

This option is incorrect

Option 3)

discontinuous everywhere

This option is incorrect

Option 4)

continuous as well as differentiable for all x

This option is incorrect

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