Question

Use Rolle's Theorem to prove that f(x) = 3x^5 + 10x^3 + 15x + 2 has at most one real root.

Answer 1

Suppose the polynomial has two or more roots, ie. there are at least two points a, b, such that f(a)=f(b)=0. Then by Rolle's theorem there is a c in (a,b) for which the derivative f'(c)=0. The derivative is as follows\[f'(x)=15x^4+30x^2+15\]Clearly, this expression is positive for all x, ie. there is no such c. This is a contradiction, hence there must be at most 1 (real) root.