The function y = ||x| - 1| is differentiable for all values of ' x ' except

Step-by-step solution

The given function is y = ||x| - 1| . A function of the form |f(x)| is generally non-differentiable at points where f(x) = 0 (if the curve crosses the x-axis) and at points where f(x) itself is non-differentiable. First, the inner function f(x) = |x| - 1 involves |x| , which is non-differentiable at x = 0 . For x in the neighborhood of 0 (i.e., -1 The left-hand derivative at x = 0 is 1 and the right-hand derivative is -1 . Thus, y is non-differentiable at x = 0 . Second, we check where the expression inside the out