The set of all points, where the function f(x) = x/(1+|x|) is differentiable, is

The set of all points, where the function f(x) = x/(1+|x|)… — JEE Main 1987 Mathematics PYQ

1987

The set of all points, where the function

f(x) = \dfrac{x}{1+|x|}

is differentiable, is

A.
$(-\infty, \infty)$
B.
$[0, \infty)$
C.
$(-\infty, 0) \cup (0, \infty)$
D.
$(0, \infty)$

Show answer and solution

Correct option: A

Solution

For x > 0, f(x) = x/(1+x), and for x < 0, f(x) = x/(1-x), both rational functions differentiable on their domains. At x = 0, compute the limit: [f(h)-f(0)]/h = [h/(1+|h|)]/h = 1/(1+|h|) → 1 as h → 0 from either side. So f'(0) = 1 exists. Therefore f is differentiable on all of (-∞, ∞).

[1987]

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JEE Main 1987 Mathematics question on Limits Continuity and Differentiability