Let f(x) = [ a x^2 - x^3 ] + | x - [x] | , where a is an integer and [t] denotes the greatest integer less than or equal to t . If f(x) is continuous at x = 2 , then the number of possible integer values of a in the interval [1, 10] is

Step-by-step solution

Given f(x) = [ a x^2 - x^3 ] + | x - [x] | . We know that x - [x] = x , which is the fractional part of x . Since x 0 , we have | x - [x] | = x . Thus, f(x) = [ a x^2 - x^3 ] + x . Let g(x) = a x^2 - x^3 . At x = 2 , g(2) = 4a - 8 , which is an integer because a is an integer. The value of the function at x = 2 is f(2) = g(2) + 0 = 4a - 8 . For f(x) to be continuous at x = 2 , we must have _ x 2^- f(x) = _ x 2^+ f(x) = f(2) . Left-Hand Limit (LHL): As x 2^- , x 1 . _ x 2^- f(x) = _ x 2^- [g(x)] + 1 . For LHL to equ