IVT Statement: If f is continuous on [a,b] and f(a)*f(b) < 0, then there exists c in (a,b) such that f(c) = 0.

Applications:

1. Proving roots exist: Show f(a) > 0 and f(b) < 0 for a continuous f.
2. Fixed point theorem: If f:[0,1]->[0,1] is continuous, then f(c) = c for some c. (Consider g(x) = f(x) - x.)
3. Showing equations have solutions: $e^x$ = 3x has a solution because g(x) = $e^x$ - 3x satisfies g(0) = 1 > 0 and g(1) = e - 3 < 0.

Important: IVT guarantees existence, not uniqueness. Multiple roots may exist.