# Misc 18 - Chapter 5 Class 12 Continuity and Differentiability

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

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 = 𝑥3 , 𝑥≥0 −𝑥3 , 𝑥<0 = 𝑥3 , 𝑥≥0 −𝑥3 , 𝑥<0 Case 1 :- When 𝑥≥0 𝑓 (𝑥)= 𝑥3 Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑓′(𝑥)= 3𝑥2 Again Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑓′′(𝑥)= 3 𝑥2′ 𝑓′′(𝑥)= 6𝑥 Hence 𝑓′′(𝑥)=6𝑥 exist for all value of 𝑥 greater than 0. Case 2 :- When 𝑥<0 𝑓 (𝑥)= −𝑥3 Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑓′ 𝑥= −𝑥3′ 𝑓′(𝑥)= −3𝑥2 Again Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑓′′(𝑥)= −3𝑥2′ 𝑓′′ 𝑥= −6𝑥 Hence 𝑓′′ 𝑥=−6𝑥 exist for all value of 𝑥 less than 0 .

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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.