using rolle's theorem to show that x^3+2x+c=0, c a constant cannot have more than one real root?

Question

please give me full answer..tq!

Answer 1

Let f(x) = x^3+2x+c.
f'(x) = 3x^2+2 is always greater than zero. If you can find at least two points such that f(c1) = f(c2) = 0, then by Rolle's theorem, you have a point c between c1 and c2 such that f'(c) = 0. This is contradictary to f'(x) is always greater than zero.Therefore, you can not find two points c1 and c2 such that f(c1) = f(c2) = 0.