Statement: If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) with f'(c) = 0.

Advanced Applications:

1. Between any two roots of f(x) = 0, there is at least one root of f'(x) = 0.
2. If f has n roots, f' has at least n-1 roots, f'' has at least n-2 roots, etc.
3. Proving uniqueness of roots: if f' > 0 everywhere, f can have at most one root.

Example: Prove $x^3$ + 3x + 1 = 0 has exactly one real root. f'(x) = 3$x^2$ + 3 > 0 always, so f is strictly increasing. f(-1) = -3 < 0, f(0) = 1 > 0. By IVT, exactly one root in (-1, 0).