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

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

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