Rolle's Applications:

1. Root counting: n roots of f => at least n-1 roots of f' => at least n-2 of f'' => ...
2. Uniqueness: if f' > 0 everywhere, f has at most one root (monotonicity argument)
3. Proving derivative equations: between roots of $e^x$ sin x, there's a root of $e^x$(sin x + cos x)

LMVT Applications:

1. Inequalities: |sin a - sin b| <= |a-b| (since |cos c| <= 1)
2. Bounds: $\frac{x}{1+x}$ < ln(1+x) < x for x > 0
3. Estimation: sqrt(26) - 5 = $\frac{1}{2sqrt(c}$) < 0.1 for c > 25

Uniqueness of Roots Strategy: To show f(x) = 0 has exactly one root:

1. Use IVT to show at least one root (find sign change)
2. Use f' > 0 (or < 0) to show at most one root (monotonicity)
3. Conclude: exactly one root

Multiple Roots Strategy: For f(x) = 0 having exactly n roots:

1. Find n sign changes (gives at least n roots)
2. Use degree argument or Rolle's to limit the count