Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12

Misc 18 - If f(x) = |x|3, show that f(x) exists and find it - Finding second order derivatives - Normal form

Slide9.JPG Slide10.JPG Slide11.JPG Slide12.JPG

  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise


Misc 18 If 𝑓 (𝑥)=|𝑥 ﷯﷮3﷯, show that 𝑓 ″(𝑥) exists for all real 𝑥 and find it. We know that 𝑥﷯= 𝑥 𝑥≥0﷮−𝑥 𝑥<0﷯﷯ 𝑓 (𝑥)=|𝑥 ﷯﷮3﷯ = 𝑥﷯﷮3﷯ , 𝑥≥0﷮ −𝑥﷯﷮3﷯ , 𝑥<0﷯﷯ = 𝑥﷮3﷯ , 𝑥≥0﷮ −𝑥﷮3﷯ , 𝑥<0﷯﷯ Case 1 :- When 𝑥≥0 𝑓 (𝑥)= 𝑥﷮3﷯ Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑓′(𝑥)= 3𝑥﷮2﷯ Again Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑓′′(𝑥)= 3 𝑥﷮2﷯﷯﷯﷮′﷯ 𝑓′′(𝑥)= 6𝑥 Hence 𝑓′′(𝑥)=6𝑥 exist for all value of 𝑥 greater than 0. Case 2 :- When 𝑥<0 𝑓 (𝑥)= −𝑥﷮3﷯ Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑓﷮′﷯ 𝑥﷯= −𝑥﷮3﷯﷯′ 𝑓′(𝑥)= −3𝑥﷮2﷯ Again Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑓′′(𝑥)= −3𝑥﷮2﷯﷯﷮′﷯ 𝑓﷮′′﷯ 𝑥﷯= −6𝑥 Hence 𝑓﷮′′﷯ 𝑥﷯=−6𝑥 exist for all value of 𝑥 less than 0 .

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.