Problem : For which values of x does the function f (x), with graph shown below, fail to be continuous? differentiable?The graph is not continuous at x = 4 and not differentiable at x = 1, 4, 5.
Problem : For which values of x does the function f (x) with the following graph have f'(x) = 0?The derivative f'(x) = 0 for x = - 2, 0. Note that the derivative is not defined at x = 2, because there is no well-defined tangent line to the graph at that point.
Problem : Let f (x) = x1/3, a function defined on the entire real line. Is f differentiable at x = 0?No, the tangent to the graph at x = 0 is vertical, with undefined slope.
Problem : Let f (x) = sin(x). For which values of x is f'(x) = 0?We have f'(x) if and only if the tangent line to the graph of f (x) at x is a horizontal line. This occurs whenever sin(x) = ±1. Thus f'(x) = 0 for x = Π/2 + kΠ, where k is any integer.