Let f(x) = x 1 + |x| be differentiable at . . . . - Vidyadip

MCQ

Let $f(x) = \frac{x}{{1 + |x|}}$ be differentiable at . . . .

✓
$( - \infty ,\infty )$
B
$[0,\infty ]$
C
$( - \infty ,\,0) \cup (0,\infty )$
D
$(0,\infty )$

✓

Answer

Correct option: A.

$( - \infty ,\infty )$

a
$f(x)=\left\{\begin{array}{ll}{\frac{x}{1-x},} & {x<0} \\ {\frac{x}{1+x},} & {x \geq 0}\end{array}\right.$

$\Rightarrow f^{\prime}(x)=\left\{\begin{array}{ll}{\frac{x}{(1-x)^{2}},} & {x<0} \\ {\frac{x}{(1+x)^{2}}} & {x \geq 0}\end{array}\right.$

$\because f^{\prime}(x)$ exist at everywhere.

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