Class 12 Continuity And Differentiability · Long Answer3 Marksdifferentiability
Class 12 Maths · Continuity And Differentiability · Long Answer #127
Show f(x)=|x−1|+|x+1| is continuous on R but not differentiable at −1 or 1.
✅ Step-by-Step Solution
Final Answer: Continuous on R; nondifferentiable exactly at −1 and 1.
- The piecewise formula is −2x for x<−1, 2 for −1≤x≤1, and 2x for x>1.
- At both junctions the adjacent values match 2; the pieces are linear elsewhere, so continuity holds on R.
- At −1 the slopes are −2 and 0; at 1 they are 0 and 2. Thus neither junction is differentiable, while all other points are.
- The claimed continuity and exact nondifferentiability set follow.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The piecewise formula is −2x for x<−1, 2 for −1≤x≤1, and 2x for x>1.
- [1 mark] At both junctions the adjacent values match 2; the pieces are linear elsewhere, so continuity holds on R.
- [1 mark] At −1 the slopes are −2 and 0; at 1 they are 0 and 2. Thus neither junction is differentiable, while all other points are.
- [0.5 marks] The claimed continuity and exact nondifferentiability set follow.
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