For real numbers a and b , consider the function f : R R given by f(x) = cases -ax - b & if x 1. cases How many pairs (a, b) are there for which f is continuous at every point of R ?

Step-by-step solution

For f(x) to be continuous at x = -1 , the left-hand limit must be equal to f(-1) . _ x -1^- f(x) = _ x -1^- (-ax - b) = a - b f(-1) = 5(-1) + 1 = -4 Equating them, we get a - b = -4 b = a + 4 . For f(x) to be continuous at x = 1 , the right-hand limit must be equal to f(1) . _ x 1^+ f(x) = _ x 1^+ (a^2x + 3b) = a^2 + 3b f(1) = 5(1) + 1 = 6 Equating them, we get a^2 + 3b = 6 . Substituting b = a + 4 into the second equation: a^2 + 3(a + 4) = 6 a^2 + 3a + 12 = 6 a^2 + 3a + 6 = 0 The discriminant of this quadratic equ