Let f : R R be defined as f(x) = [e^ x ] for x < 0; ae^ x + [x - 1] for 0 x < 1; b + [ ( x)] for 1 x < 2; [e^ -x ] - c for x 2. Where a, b, c R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?

JEE Main 2022 Mathematics question

Let f : be defined as for x < 0; for ; for ; for . Where a, b, c R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?

(1)There exists a, b, c R such that f is continuous on R
(2)If f is discontinuous at exactly one point, then a + b + c = 1
(3)If f is discontinuous at exactly one point, then Correct
(4)f is discontinuous at atleast two points, for any values of a, b and c

Answer: (3)

Solution

Simplifying: f(x) = 0 for x < 0; for ; f(x) = b for x = 1 and b − 1 for 1 < x < 2; and f(x) = −c for . Continuity at x = 0 needs a − 1 = 0, so a = 1. Continuity at x = 1 needs ae − 1 =

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Question ID VK-034963