Class 12 Continuity And Differentiability · Short Answer2 Marksdifferentiability
Class 12 Maths · Continuity And Differentiability · Short Answer #51
Let f(x)=x² for x≥1 and f(x)=x for x<1. Show that f is not differentiable at 1.
✅ Step-by-Step Solution
Final Answer: Continuous but not differentiable at 1.
- At 1, both branch limits and f(1) equal 1, so f is continuous.
- For h>0 the left quotient [(1−h)−1]/(−h)=1, while the right quotient [(1+h)²−1]/h=2+h→2.
- The one-sided derivatives 1 and 2 differ, so f is not differentiable at 1.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] At 1, both branch limits and f(1) equal 1, so f is continuous.
- [1 mark] For h>0 the left quotient [(1−h)−1]/(−h)=1, while the right quotient [(1+h)²−1]/h=2+h→2.
- [0.5 marks] The one-sided derivatives 1 and 2 differ, so f is not differentiable at 1.
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