Mean value theorems
Objective. Apply Rolle and mean-value theorems rigorously.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Mean value theorems relate a function's average rate of change to its instantaneous derivative. Rolle's theorem states that if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then f'(c) = 0 for some c in (a, b). The Mean Value Theorem generalises this: f'(c) = (f(b) − f(a))/(b − a) for some c ∈ (a, b). For f(x) = x² on [1, 3], the average rate is (9 − 1)/2 = 4, and f'(c) = 2c = 4 gives c = 2, which indeed lies in (1, 3). These theorems justify many results about monotonicity and bounding function values.
Worked example
Use the Mean Value Theorem to find a value c guaranteed for f(x) = x³ on the interval [0, 2].
- 1Check f is continuous on [0, 2] and differentiable on (0, 2), so the theorem applies.
- 2Compute the average rate of change: (f(2) − f(0))/(2 − 0) = (8 − 0)/2 = 4.
- 3Differentiate: f'(x) = 3x², so set 3c² = 4 giving c² = 4/3.
- 4Taking the positive root in (0, 2) gives c = 2/√3.
Answer: c = 2/√3
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Start practisingCommon mistake. Students forget to check differentiability on the open interval; applying the Mean Value Theorem to f(x) = |x| on [−1, 1] is invalid since f is not differentiable at 0, so no valid conclusion about c follows.
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