NCERT Class 12 · Ch 5 Miscellaneous Exercise · Q18
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Miscellaneous Exercise Question 18
For f(x)=|x|³, show f″ exists for every real x and find it.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: f″(x)=6|x|, including f″(0)=0.
- For x>0, f=x³, f′=3x² and f″=6x. For x<0, f=−x³, f′=−3x² and f″=−6x.
- At 0, f′(0)=lim |h|³/h=0. Also f′(h)=3h|h|, so f″(0)=lim 3h|h|/h=lim 3|h|=0.
- These cases combine into f″(x)=6|x| on R.
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