NDA 2024 · PYQ · Continuity and Differentiability · medium
Let f(x) = |x| + 1 and g(x) = [x] - 1, where [.] is the greatest integer function. Let h(x) = f(x)/g(x). Consider the following statements: 1. f(x) is differentiable for all x < 0. 2. g(x) is continuous at x = 0.0001. 3. The derivative of g(x) at x = 2.5 is 1. Which of the statements given above are correct?
A.1 and 2 only✓ Correct
B.2 and 3 only
C.1 and 3 only
D.1, 2 and 3
Explanation
Statement 1: f(x) = |x| + 1 = -x + 1 for x < 0, which is linear hence differentiable. True. Statement 2: g(x) = [x] - 1. At x = 0.0001, [0.0001] = 0, and g is constant = -1 in a neighborhood (0, 1), so g is continuous at x = 0.0001. True. Statement 3: At x = 2.5, [x] = 2 in a neighborhood, so g(x) = 1 (constant). The derivative is 0, not 1. False. Hence statements 1 and 2 only are correct.
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