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Class 12 Continuity And Differentiability · Long Answer3 Marksdifferentiability

Class 12 Maths · Continuity And Differentiability · Long Answer #121

Let f(x)=3x−2 for 0<x≤1, 2x²−x for 1<x≤2, and 5x−4 for x>2. Show f is continuous but not differentiable at 2.

✅ Step-by-Step Solution

Final Answer: Continuous at 2; left derivative 7, right derivative 5.
  1. The value f(2)=2·4−2=6.
  2. The left limit of 2x²−x is 6 and the right limit of 5x−4 is 6, proving continuity at 2.
  3. The one-sided derivatives are (4x−1)|₂=7 and 5; their difference shows no ordinary derivative exists.
  4. Thus f is continuous at 2 but not differentiable there.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] The value f(2)=2·4−2=6.
  • [1 mark] The left limit of 2x²−x is 6 and the right limit of 5x−4 is 6, proving continuity at 2.
  • [1 mark] The one-sided derivatives are (4x−1)|₂=7 and 5; their difference shows no ordinary derivative exists.
  • [0.5 marks] Thus f is continuous at 2 but not differentiable there.

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