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Let f: R R be a function defined as f(x) = cases e^ x+2 -1 x+2 + a & if x 2 cases where [t] denotes the greatest integer t . If f(x) has exactly 9 points of discontinuity on R , then the value of a + b is ______

Correct answer

2

Step-by-step solution

For x (-2, 2) , the function is f(x) = [ 6 x^2 + 1 ] . Let g(x) = 6 x^2 + 1 . For x (-2, 2) , g(x) is continuous, strictly increasing on (-2, 0] , strictly decreasing on [0, 2) , and its range is (1.2, 6] . The function [g(x)] is discontinuous at points where g(x) crosses an integer. In (-2, 2) , g(x) takes the integer values 2, 3, 4, 5 twice each, giving 8 points of discontinuity. At x = 0 , g(x) reaches a local maximum of 6 . Here, _ x 0 [ 6 x^2 + 1 ] = 5 (since g(x) Total points of discontinuity in (-2, 2) are 8

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