Let f(x) = [ k 1+x^2 ] , where x (0, 2) , k is a positive integer and [t] denotes the greatest integer less than or equal to t . If f(x) is not differentiable at exactly 15 points in (0, 2) , then the sum of all possible values of k is

Step-by-step solution

Let g(x) = k 1+x^2 . On the interval (0, 2) , g(x) is strictly decreasing. The range of g(x) for x (0, 2) is ( k 5 , k ) . Since g(x) is strictly monotonic, f(x) = [g(x)] is discontinuous (and hence non-differentiable) exactly at the points where g(x) takes an integer value. The number of integers strictly between k 5 and k is given by (k-1) - [ k 5 ] . We are given that this number is 15 , so: (k-1) - [ k 5 ] = 15 k - [ k 5 ] = 16 Let us test integer values for k : If k = 18 , 18 - [ 18 5 ] = 18 - 3 = 15 (Rejected