Question 18
Let f (x) = |sin x|. Then
(A) f is everywhere differentiable
(B) f is everywhere continuous but not differentiable at x = nš, nā Z.
(C) f is everywhere continuous but not differentiable at x = (2n + 1) š/2, nā Z.
(d) None of these
f(š„) = |sin š„|
We need to check continuity and differentiability of f(š„)
Continuity of f(š)
Let š(š„)=|š„| & ā(š„)=sinā”š„
Then,
ššš(š)=š(ā(š„))
=š(sinā”š„ )
=|sinā”š„ |
=š(š)
ā“ š(š„)=ššā(š„)
We know that,
š(š)=š¬š¢š§ā”š is continuous as sin is continuous
š(š)=|š| is continuous as it is a modulus function
Hence, g(š„) & h(š„) both are continuous
And
If two functions g(š„) & h(š„) are continuous then their composition ššā(š„) is also continuous
ā“ š(š) is continuous
Differentiability of š(š)
š(š„)=|sinā”š„ |
Since, it is a modulus function so we check differentiability when sinā”š„=0
i.e., š=šš , šāš
š(š„) is differentiable at š„=šš, if
LHD = RHD
(ššš)ā¬(š”āš) (š(š) ā š(š ā š))/š
= (ššš)ā¬(hā0) (š(šš) ā š(ššā ā))/ā
= (ššš)ā¬(hā0) (|sinā”šš | ā|sinā”ć(šš ā ā)ć |)/ā
= (ššš)ā¬(hā0) (|0| ā |sinā”ć(šš ā ā)ć |)/ā
Using sin (A ā B) = sin Acos B ā cos Asin B
= (ššš)ā¬(hā0) (0 ā |sinā”ćšš cosā”ćā ā cosā”ćšš sinā”ā ć ć ć |)/ā
= (ššš)ā¬(hā0) ( ā |0 ā cosā”ćšš sinā”ā ć |)/ā
= (ššš)ā¬(hā0) ( ācosā”ćšš sinā”ā ć)/ā
= ācos šš Ć(ššš)ā¬(hā0) sinā”ā/ā
Using (ššš)ā¬(š„ā0) š ššā”š„/š„=1
=ācos šš Ć1
=āššØš¬ šš
(ššš)ā¬(š”āš) (š(š + š) ā š(š ))/š
= (ššš)ā¬(hā0) (š(šš+ ā) ā š(šš))/ā
= (ššš)ā¬(hā0) (|sinā”ć(šš+ā)ć |ā|sinā”šš |)/ā
= (ššš)ā¬(hā0) (|sinā”ć(šš+ā)ć |ā|0|)/( ā)
Using sin (A + B) = sin Acos B + cos Asin B
= (ššš)ā¬(hā0) (|sinā”ćšš cosā”ćā + cosā”ćšš sinā”ā ć ć ć | ā 0)/ā
= (ššš)ā¬(hā0) |0 + cosā”ćšš sinā”ā ć |/ā
= (ššš)ā¬(hā0) cosā”ćšš sinā”ā ć/ā
= cos šš Ć(ššš)ā¬(hā0) sinā”ā/ā
Using (ššš)ā¬(š„ā0) š ššā”š„/š„=1
= cos šš Ć1
= ššØš¬ šš
Since,
LHD ā RHD
ā“ š(š„) is not differentiable at š=šš
Hence, š(š„) is continuous everywhere but not differentiable at šāšš , šāš
So, the correct answer is (B)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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