# Let $f:R\rightarrow R$ be a function defined by $f\left ( x \right )= min\left \{ x+1,\left | x \right |+1 \right \}$Then which of the following is true ? Option 1) $f\left ( x \right )$ is differentiable everywhere Option 2) $f\left ( x \right )$ is not differentiable at $x= 0$ Option 3) $f\left ( x \right )$ $\geq 1\: for \: all\: x\in R$ Option 4) $f\left ( x \right )$ is not differentiable at $x= 1$

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

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$f(x)=\:min\begin{Bmatrix} x+1,\left | x \right |+1 \end{Bmatrix}$

$f(x)= x+1, x\:\epsilon R$

So f(x) is differentiable every where.

Option 1)

$f\left ( x \right )$ is differentiable everywhere

Correct

Option 2)

$f\left ( x \right )$ is not differentiable at $x= 0$

Incorrect

Option 3)

$f\left ( x \right )$ $\geq 1\: for \: all\: x\in R$

Incorrect

Option 4)

$f\left ( x \right )$ is not differentiable at $x= 1$

Incorrect

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