Limits, Continuity & Differentiability
Continuity and Differentiability
nta_pyq_2025_jan
Grade 12
Question:
⎧ ⎪ 3x, x < 0 Let f (x) = ⎨ min{1 + x + [x], x + 2[x]}, 0 \le x \le 2 ⎩ ⎪ 5, x > 2, where [.] denotes greatest integer function. If \alpha and \beta are the number of points, where f is not continuous and is not differentiable, respectively, then \alpha + \beta equals __________
Step-by-Step Solution
Key Concept: Apply the core result for continuity and differentiability at a point and simplify using the given constraints.
⎧ ⎪ ⎪ 3x ; x < 0 ⎪ ⎪ (5) f (x) = ⎨ min{1 + x, x} ; 0 \le x < 1 ⎪ min{2 + x, x + 2} ; 1 \le x < 2 ⎪ ⎪ ⎩ ⎪ 5 ; x > 2 ⎧ 3x ; x < 0 ⎪ ⎪ ⎪ ⎪ x ; 0 \le x < 1 f (x) = ⎨ ⎪ x + 2 ; 1 \le x < 2 ⎪ ⎪ ⎪ ⎩ 5 ; x > 2 Not continuous at x \in {1, 2} \Rightarrow \alpha = 2 Not diff. at x \in {0, 1, 2} \Rightarrow \beta = 3 \alpha + \beta = 5
Correct Answer: 5