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Let [t] denote the greatest integer less than or equal to t . Consider the function f(x) defined as: f(x) = array cl 1- (ax) 2x^2 , & x 0 array . If f(x) is continuous at x=0 , then the number of ordered pairs of integers (a, b) that satisfy this condition is

Options

  1. A0
  2. B2
  3. C4
  4. D1

Correct answer

C. 4

Step-by-step solution

For f(x) to be continuous at x=0 , we must have _ x 0^- f(x) = _ x 0^+ f(x) = f(0) . Given f(0) = 9 . Evaluating the Left Hand Limit (LHL): _ x 0^- 1- (ax) 2x^2 = _ x 0^- 2 ^2 ( ax 2 ) 2x^2 = _ x 0^- ^2 ( ax 2 ) ( ax 2 )^2 a^2 4 = a^2 4 . Equating LHL to f(0) : a^2 4 = 9 a^2 = 36 a = 6 . Evaluating the Right Hand Limit (RHL): As x 0^+ , x approaches 1 from values less than 1 (i.e., 0 Therefore, the greatest integer [ x] = 0 . _ x 0^+ b^2 ( 2 [ x] ) = b^2 ( 2 0 ) = b^2 (0) = b^2 . Equating RHL to f(0) : b^2 = 9 b =

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