Which of the following function is differentiable for all x- where - represents the greatest integer function and -sgn- represents signum function-

Name: JEE- Grade 12- Mathematics- Continuity and Differentiability- Session 05- D01
Uploaded: 2022-07-04T10:36:42+05:30
Description: |x| is not differentiable at x=0 ( because at x=0, there is sharp corner.) [x] is not continuous for all x∈ℤ Hence, it will not differentiable for all x∈ℝ x^2 ...

|x| is not differentiable at x=0 ( because at x=0, there is sharp corner.) [x] is not continuous for all x∈ℤ Hence, it will not differentiable for all x∈ℝ x^2 ...

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Standard XI

Mathematics

Continuity of a Function

Which of the ...

Question

Which of the following function is differentiable for all $x \in R$ (where $[.]$ represents the greatest integer function and $^{'} {sgn}^{'}$ represents signum function.)

| x |

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[x]

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x^{2} + 2 x + 1

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sgn x

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Solution

The correct option is C $x^{2} + 2 x + 1$
$| x |$ is not differentiable at $x = 0$
( because at $x = 0,$ there is sharp corner.)

$[x]$ is not continuous for all $x \in Z$
Hence, it will not differentiable for all $x \in R$

$x^{2} + 2 x + 1$ is continuous as well as differentiable for all $x \in R$

$sgn x$ is not continuous at $x = 0$
Hence, it is not differentiable for all $x \in R$

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Continuity of a Function

Standard XI Mathematics

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Which of the following function is differentiable for all x∈R (where [.] represents the greatest integer function and ′sgn′ represents signum function.)

Which of the following function is differentiable for all $x \in R$ (where $[.]$ represents the greatest integer function and $^{'} {sgn}^{'}$ represents signum function.)