Let g(x) = |x^2 - 25| - 10|x - 5| + [ x 2 ] - [ x 3 ] , where [t] denotes the greatest integer less than or equal to t . The number of points in the open interval (-20, 20) at which g(x) is not differentiable is ______.

Step-by-step solution

Let g(x) = M(x) + G(x) , where M(x) = |x^2 - 25| - 10|x - 5| and G(x) = [ x 2 ] - [ x 3 ] . First, analyze M(x) at its critical points x = 5 and x = -5 . At x = 5 : For x 5^+ , M(x) = (x^2 - 25) - 10(x - 5) = x^2 - 10x + 25 = (x - 5)^2 . The right-hand derivative is 2(x - 5) 0 . For x 5^- , M(x) = -(x^2 - 25) - 10(-(x - 5)) = -x^2 + 10x - 25 = -(x - 5)^2 . The left-hand derivative is -2(x - 5) 0 . Since both derivatives are 0 , M(x) is differentiable at x = 5 . At x = -5 : For x -5^+ , M(x) = -(x^2 - 25) - 10(-(x -