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NDA 2024 Maths · Mathematics · Derivatives · hard

If f(x)=|x|^3, then f''(0) is equal to

Single correct option

If f(x)=∣x∣3f(x)=|x|^3, then f′′(0)f''(0) is equal to
  1. A. 66
  2. B. 33
  3. C. 00
  4. D. does not exist
Show answer and solution

Answer: (C) 00

For x>0x>0, f=x3f=x^3 so f′=3x2f'=3x^2 and f′′=6xf''=6x; for x<0x<0, f=−x3f=-x^3 so f′=−3x2f'=-3x^2 and f′′=−6xf''=-6x. In both cases f′(x)=3x∣x∣f'(x)=3x|x| and f′′(x)=6∣x∣f''(x)=6|x| which tends to 00 at x=0x=0; the second derivative at 0 from the definition is lim⁡3h∣h∣h=0\lim\frac{3h|h|}{h}=0. Trap: assuming the modulus function always produces a corner.
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