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CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q364 Markspiecewise_continuity_differentiability

CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Question 36 (Continuity And Differentiability)

A racing track has equation y=f(x), where f(x)=x⁴−4x²+4 for 0≤x<3 and f(x)=x²+40 for x≥3.
Indicative racing circuit illustration. Use the piecewise formula to answer the questions.
Indicative racing circuit illustration. Use the piecewise formula to answer the questions.
Part (1) 1 Marks
Find f′(x) for 0<x<3.
Answer: 4x³−8x.
  1. For 0<x<3, use f(x)=x⁴−4x²+4.
  2. Differentiate term by term to obtain 4x³−8x.
Part (2) 1 Marks
Find f′(4).
Answer: 8.
  1. For x>3, differentiate x²+40 to get 2x.
  2. Substitute x=4 to get 8.
Part (3) 2 Marks
Test continuity at x=3.
Answer: Continuous at x=3.
  1. Use the lower polynomial for the left-hand limit and the upper polynomial for the right-hand limit.
  2. The left limit is 81−36+4=49. The right limit and f(3) are 9+40=49.
  3. The two limits and the function value agree, so f is continuous there.
Part (4) 2 Marks
Test differentiability at x=3.
Answer: Not differentiable at x=3.
  1. Both polynomials extend to the same value f(3)=49, so each one-sided derivative equals that polynomial’s derivative at 3.
  2. The left derivative is 4·27−8·3=84, and the right derivative is 2·3=6.
  3. The one-sided derivatives differ, so f′(3) does not exist.

✅ Step-by-Step Solution

Review the worked subparts and marking breakdown above.

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All Questions in CBSE Class 12 Mathematics 2026 Main Paper 65/2/3

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