Let S be the set of points in the open interval (-20, 20) at which the function f(x) = |x^2 - 6x| + [ x a ] is not differentiable, where a is a positive integer and [t] denotes the greatest integer less than or equal to t . If the number of elements in S is exactly 6 , then the sum of all possible values of a is ______.

Step-by-step solution

The given function is f(x) = |x^2 - 6x| + [ x a ] . First, consider the modulus part M(x) = |x^2 - 6x| . The roots of x^2 - 6x = 0 are x = 0 and x = 6 . Since the derivative of x^2 - 6x is non-zero at these points, M(x) is non-differentiable at x = 0 and x = 6 . Next, consider the greatest integer function part G(x) = [ x a ] . This function is discontinuous, and hence non-differentiable, at all points where x a is an integer. Thus, G(x) is non-differentiable at x = ka for any integer k . In the interval (-20, 20)