Last updated at May 6, 2021 by Teachoo

Transcript

Misc 18 If π (π₯)=|π₯|^3, show that π β³(π₯) exists for all real π₯ and find it. We know that |π₯|={β( π₯ π₯β₯0@βπ₯ π₯<0)β€ Therefore, π (π₯)=|π₯|^3 = {β( (π₯)^3 , π₯β₯0@(βπ₯)^3 , π₯<0)β€ = {β( π₯^3 , π₯β₯0@γβπ₯γ^3 , π₯<0)β€ Case 1: When πβ₯π π (π₯)=π₯^3 Differentiating π€.π.π‘.π₯. πβ²(π₯)=γ3π₯γ^2 Again Differentiating π€.π.π‘.π₯. πβ²β²(π₯)= (3π₯^2 )^β² πβ²β²(π)=" " 6π₯ Hence, πβ²β²(π) exists for all value of π₯ greater than 0. Case 2: When π<π π (π₯)=γβπ₯γ^3 Differentiating π€.π.π‘.π₯. πβ²(π₯)=γβ3π₯γ^2 Again Differentiating π€.π.π‘.π₯. πβ²β²(π₯)= (γβ3π₯γ^2 )^β² π^β²β² (π)=" "β6π₯ Hence, πβ²β²(π) exists for all value of π₯ less than 0. Case 3: At x = 0 To check if πβ²β²(π) exists for x = 0, We need to check differentiability of πβ²(π) at π = π Here, π(π₯)= {β( π₯^3 , π₯β₯0@γβπ₯γ^3 , π₯<0)β€ πβ²(π)= {β( γ3π₯γ^2 , π₯β₯0@γβ3π₯γ^2 , π₯<0)β€ We know that πβ²(π₯) is differentiate at π₯ = 0 if L.H.D = R.H.D(π₯π’π¦)β¬(π βπ ) (π^β² (π) β π^β² (π β π))/π = limβ¬(β β0 ) (π^β² (0) β π^β² (ββ))/β = limβ¬(β β0 ) (γ3(0)γ^2 β(βγ3(ββ)γ^2))/β = limβ¬(β β0 ) γ3βγ^2/β = limβ¬(h β0 ) (3β) Putting β =0 = 3(0) = 0 (π₯π’π¦)β¬(π βπ ) (π^β² (π + π) βπ(π))/π = limβ¬(β β0 ) (γππγ^β² (β) β π(0))/(β ) = limβ¬(β β0 ) (γ3(β)γ^2 β γ3(0)γ^2)/β = limβ¬(β β0 ) γ3βγ^2/β = limβ¬(β β0 ) 3β Putting β =0 = 3(0) = 0 Thus, LHD = RHD Therefore, π^β² (π) is differentiable at π₯ = 0 So, we can say that π^β²β² (π) exists for x = 0 a Thus, π^β²β²(π) exists for all real values of π₯ Hence proved

Miscellaneous

Misc 1

Misc 2

Misc 3

Misc 4

Misc 5 Important

Misc 6 Important

Misc 7 Important

Misc 8

Misc 9 Important

Misc 10

Misc 11 Important

Misc 12

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

Misc 17 Important

Misc 18 You are here

Misc 19 Important

Misc 20

Misc 21

Misc 22

Misc 23 Important

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.