posted by Gabe .
Use the Intermediate Value Theorem to show that there is a root in the equation x^(1/3)=1-x in the interval (0,1).
let f(x) = x^(1/3) + x - 1
Show that f(0) > 0 and f(1) < 0, or vice versa. Then because f is continuous on the interval (0, 1), f(c) must equal 0 for at least one c on the interval (0, 1).