Let f(x) = (|x|, x^2) and g(x) = |x - 1 2 | . Let m be the number of points where f(x) is not differentiable, and n be the number of points where the composite function h(x) = g(f(x)) is not differentiable. The value of m + n is:

Step-by-step solution

For f(x) = (|x|, x^2) : When x [-1, 1] , |x| x^2 f(x) = |x| . When |x| > 1 , x^2 > |x| f(x) = x^2 . The points of non-differentiability of f(x) are x = -1, 0, 1 . Thus, m = 3 . For the composite function h(x) = g(f(x)) = |f(x) - 1 2 | : h(x) is non-differentiable at points where f(x) = 1 2 (since f'(x) 0 at these points) and at points where f(x) is inherently non-differentiable (provided f(x) 1 2 at these points). Setting f(x) = 1 2 |x| = 1 2 x = 1 2 . The points where f(x) is inherently non-differentiable are x =