Class 12 Continuity And Differentiability · Long Answer3 Marksdifferentiability
Class 12 Maths · Continuity And Differentiability · Long Answer #65
Let f(x)=ax²+b for x<1 and f(x)=2x+1 for x≥1. Find a,b for differentiability at 1.
✅ Step-by-Step Solution
Final Answer: a=1,b=2
- Differentiability implies continuity, so a+b=f(1)=3.
- Under that condition the left difference quotient tends to 2a and the right quotient to 2.
- Equality of derivatives gives 2a=2, hence a=1 and b=3−1=2.
- With a=1,b=2, the values and one-sided derivatives match; these are the required constants.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Differentiability implies continuity, so a+b=f(1)=3.
- [1 mark] Under that condition the left difference quotient tends to 2a and the right quotient to 2.
- [1 mark] Equality of derivatives gives 2a=2, hence a=1 and b=3−1=2.
- [0.5 marks] With a=1,b=2, the values and one-sided derivatives match; these are the required constants.
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