Question 18
Let f (x) = |sin x|. Then
(A) f is everywhere differentiable
(B) f is everywhere continuous but not differentiable at x = nπ, nβ Z.
(C) f is everywhere continuous but not differentiable at x = (2n + 1) π/2, nβ Z.
(d) None of these
f(π₯) = |sin π₯|
We need to check continuity and differentiability of f(π₯)
Continuity of f(π)
Let π(π₯)=|π₯| & β(π₯)=sinβ‘π₯
Then,
πππ(π)=π(β(π₯))
=π(sinβ‘π₯ )
=|sinβ‘π₯ |
=π(π)
β΄ π(π₯)=ππβ(π₯)
We know that,
π(π)=π¬π’π§β‘π is continuous as sin is continuous
π(π)=|π| is continuous as it is a modulus function
Hence, g(π₯) & h(π₯) both are continuous
And
If two functions g(π₯) & h(π₯) are continuous then their composition ππβ(π₯) is also continuous
β΄ π(π) is continuous
Differentiability of π(π)
π(π₯)=|sinβ‘π₯ |
Since, it is a modulus function so we check differentiability when sinβ‘π₯=0
i.e., π=ππ , πβπ
π(π₯) is differentiable at π₯=ππ, if
LHD = RHD
(πππ)β¬(π‘βπ) (π(π) β π(π β π))/π
= (πππ)β¬(hβ0) (π(ππ) β π(ππβ β))/β
= (πππ)β¬(hβ0) (|sinβ‘ππ | β|sinβ‘γ(ππ β β)γ |)/β
= (πππ)β¬(hβ0) (|0| β |sinβ‘γ(ππ β β)γ |)/β
Using sin (A β B) = sin Acos B β cos Asin B
= (πππ)β¬(hβ0) (0 β |sinβ‘γππ cosβ‘γβ β cosβ‘γππ sinβ‘β γ γ γ |)/β
= (πππ)β¬(hβ0) ( β |0 β cosβ‘γππ sinβ‘β γ |)/β
= (πππ)β¬(hβ0) ( βcosβ‘γππ sinβ‘β γ)/β
= βcos ππ Γ(πππ)β¬(hβ0) sinβ‘β/β
Using (πππ)β¬(π₯β0) π ππβ‘π₯/π₯=1
=βcos ππ Γ1
=βππ¨π¬ ππ
(πππ)β¬(π‘βπ) (π(π + π) β π(π ))/π
= (πππ)β¬(hβ0) (π(ππ+ β) β π(ππ))/β
= (πππ)β¬(hβ0) (|sinβ‘γ(ππ+β)γ |β|sinβ‘ππ |)/β
= (πππ)β¬(hβ0) (|sinβ‘γ(ππ+β)γ |β|0|)/( β)
Using sin (A + B) = sin Acos B + cos Asin B
= (πππ)β¬(hβ0) (|sinβ‘γππ cosβ‘γβ + cosβ‘γππ sinβ‘β γ γ γ | β 0)/β
= (πππ)β¬(hβ0) |0 + cosβ‘γππ sinβ‘β γ |/β
= (πππ)β¬(hβ0) cosβ‘γππ sinβ‘β γ/β
= cos ππ Γ(πππ)β¬(hβ0) sinβ‘β/β
Using (πππ)β¬(π₯β0) π ππβ‘π₯/π₯=1
= cos ππ Γ1
= ππ¨π¬ ππ
Since,
LHD β RHD
β΄ π(π₯) is not differentiable at π=ππ
Hence, π(π₯) is continuous everywhere but not differentiable at πβππ , πβπ
So, the correct answer is (B)

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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