Class 12 Continuity And Differentiability · Long Answer3 Marksdifferentiability
Class 12 Maths · Continuity And Differentiability · Long Answer #60
Show f(x)=2x−|x| is continuous but not differentiable at 0.
✅ Step-by-Step Solution
Final Answer: Continuous at 0; left derivative 3, right derivative 1.
- For x<0, f(x)=3x; for x≥0, f(x)=x.
- The two limits at 0 are 0 and f(0)=0, proving continuity.
- The left quotient f(−h)/(−h)=3 and right quotient f(h)/h=1 for h>0, so their limits are different.
- Thus f is continuous at 0 but is not differentiable there.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] For x<0, f(x)=3x; for x≥0, f(x)=x.
- [1 mark] The two limits at 0 are 0 and f(0)=0, proving continuity.
- [1 mark] The left quotient f(−h)/(−h)=3 and right quotient f(h)/h=1 for h>0, so their limits are different.
- [0.5 marks] Thus f is continuous at 0 but is not differentiable there.
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