First slide

Differentiability

Question

$The number of points at which g (x) = \frac{1}{1 + \frac{2}{f (x)}} is not differentiable, where f (x) = \frac{1}{1 + \frac{1}{x}}$ is

Moderate

A

1
B

2
C

3
D

4

Solution

$Since f (x) = \frac{1}{1 + \frac{1}{x}}$
$= \frac{x}{x + 1} is not differentiable at x = 0, - 1$

$also g (x) = \frac{1}{1 + \frac{2}{f (x)}}$
$= \frac{1}{1 + \frac{2}{\frac{x}{x + 1}}}$
$= \frac{1}{1 + \frac{2 (x + 1)}{x}}$
$= \frac{x}{x + 2 (x + 1)}$
$= \frac{x}{3 x + 2} is not differentiable at x = 0, - 1, \frac{- 2}{3}$
$∴ g (x) is not differentiable at 3 points$

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