Applications of Derivatives · Theorems
Rolle's Theorem
If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists at least one c in (a,b) where f'(c) = 0. This is a special case of MVT.
Conditions. f continuous on [a,b], differentiable on (a,b), and f(a) = f(b).
Worked examples
Find c satisfying Rolle's theorem for f(x) = x² - 4x + 3 on [1, 3].
- f(1) = 1-4+3 = 0, f(3) = 9-12+3 = 0. So f(1) = f(3) ✓
- f'(x) = 2x - 4 = 0 → x = 2
- c = 2 is in (1, 3) ✓
Answer: c = 2
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