Let f: R R be defined as f(x) = cases a + [e^x + 1], & x where a, b, c, d R and [t] denotes the greatest integer less than or equal to t . If f is discontinuous at exactly one point, then the value of 2b - 2a + 2c - d is

Step-by-step solution

For f(x) to be continuous at x=0 : _ x 0^- f(x) = _ x 0^- (a + [e^x + 1]) = a + 1 _ x 0^+ f(x) = f(0) = b Thus, a + 1 = b b - a = 1 At x=1 : f(1) = c(1) + [ ( 2 )] = c + 1 _ x 1^+ f(x) = _ x 1^+ (cx + [ ( x 2 )]) = c + 0 = c Since f(1) _ x 1^+ f(x) , f(x) is discontinuous at x=1 for all values of c . For f(x) to be continuous at x=2 : _ x 2^- f(x) = _ x 2^- (cx + [ ( x 2 )]) = 2c + 0 = 2c _ x 2^+ f(x) = f(2) = d + [ ₂( 2 2 + 1)] = d + 1 Thus, 2c = d + 1 2c - d = 1 Since f(x) is discontinuous at exactly one point (w