Class 12 Continuity And Differentiability · Long Answer3 Marksdifferentiability
Class 12 Maths · Continuity And Differentiability · Long Answer #64
Discuss continuity and differentiability of f(x)=|x|+|x−1| on (−1,2).
✅ Step-by-Step Solution
Final Answer: Continuous throughout; nondifferentiable only at 0 and 1.
- Write f(x)=1−2x for −1<x<0, f(x)=1 for 0≤x≤1, and f(x)=2x−1 for 1<x<2.
- At 0 and 1 the adjacent pieces meet at value 1; elsewhere each piece is linear. Therefore f is continuous throughout the interval.
- At 0 the one-sided slopes are −2 and 0; at 1 they are 0 and 2. Thus differentiability fails at these two points, while each other point has a locally linear formula.
- Hence f is continuous on (−1,2) and differentiable on (−1,2)∖{0,1}.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Write f(x)=1−2x for −1<x<0, f(x)=1 for 0≤x≤1, and f(x)=2x−1 for 1<x<2.
- [1 mark] At 0 and 1 the adjacent pieces meet at value 1; elsewhere each piece is linear. Therefore f is continuous throughout the interval.
- [1 mark] At 0 the one-sided slopes are −2 and 0; at 1 they are 0 and 2. Thus differentiability fails at these two points, while each other point has a locally linear formula.
- [0.5 marks] Hence f is continuous on (−1,2) and differentiable on (−1,2)∖{0,1}.
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