Show that $\displaystyle 1+2x+x^3+4x^5=0$ has exactly one root using Rolle's Theorem.
1)Consider f ' (x) = 2 + 3x^2 + 20x^4
Using the quadratic formula there are no zeroes of the derivative
2) If f(x) had more than one zero say x =a and b then by Rolles Thm there would be a number c in(a,b) st f ' (c) = 0 which contradicts 1)
3) Since f(x) is a fifth degree polynomial it has at least one zero
Therefore combining 2) and 3) f has exactly one zero