Rolle's theorem cannot be applied to the function f(x) = x1/3 on the interval [-1, 1] because

    to use rolle's theorem for a function f on [a,b], it needs to be continuous on [a,b] and differentiable on (a,b).you can't apply it to f on [-1,1] because f is not differentiable on the closed interval (-1,1). if we find the derivative    then we see that derivative of f is not defined at x = 0 (f has a vertical tangent line there).it's not differentiable at x = 0, so it's not differentiable on (-1,1); therefore, we cannot apply rolle's theorem for f on [-1,1].