Mean Value Theorems

Mean value theorems connect average and instantaneous rates. Learn Rolle's theorem, Lagrange's mean value theorem and Cauchy's generalised form, their assumptions, geometric interpretation, and applications to bounds, error estimates, uniqueness of roots and speed enforcement.

If a car averages 90 km/h over a trip, at some instant its speedometer must read exactly 90 km/h. This intuitive fact is the mean value theorem — and it underlies Taylor error bounds, numerical analysis and L'Hôpital's rule.

An analogy

Average-speed cameras rely on the MVT: if your average speed between two cameras exceeds the limit, you must have exceeded it at some moment.

Assumptions

f must be continuous on the closed interval and differentiable on the open interval. |x| on [−1, 1] fails (corner at 0), so Rolle's conclusion fails there.

Geometric meaning

Somewhere between a and b the tangent line is parallel to the secant (chord) joining the endpoints. Rolle's theorem is the special case of a horizontal chord.

Uses

Proving inequalities (|sin a − sin b| ≤ |a − b|), showing an equation has at most one root (if f′ ≠ 0), bounding errors of approximations, and deriving Taylor's remainder.

Cauchy form

Relates two functions' changes; it is the key step in proving L'Hôpital's rule and generalises to parametric curves.

Formulas

• Rolle: f(a) = f(b) ⇒ ∃c ∈ (a, b): f′(c) = 0
• Lagrange MVT: ∃c ∈ (a, b): f′(c) = (f(b) − f(a))/(b − a)
• Cauchy MVT: (f(b) − f(a)) g′(c) = (g(b) − g(a)) f′(c)
• Bound: |f(b) − f(a)| ≤ max|f′| · |b − a|

Worked example: Find c

Given: f(x) = x³ on [0, 2]

Find: c in the MVT.

• Average slope = 8/2 = 4
• 3c² = 4 → c = 2/√3 = 1.155

Answer: c ≈ 1.155 ∈ (0, 2)

Worked example: Uniqueness of a root

Given: f(x) = x³ + 2x − 5

Find: How many real roots?

• f′ = 3x² + 2 > 0 → strictly increasing
• f(1) = −2, f(2) = 7 → one root (IVT)
• Two roots would need f′ = 0 somewhere (Rolle) — impossible

Answer: Exactly one real root

Common mistakes

• Wrong: Applying MVT to non-differentiable functions. — Right: Check corners and cusps.
• Wrong: Thinking c is unique. — Right: There may be several such points.

Applications

• Average-speed enforcement: Legal basis of section speed control.
• Error bounds: Taylor remainder and numerical method errors.
• Uniqueness proofs: Monotone functions have at most one root.

Key takeaways

• Continuous on [a, b], differentiable on (a, b).
• f′(c) = average slope.
• Rolle is the horizontal case.
• Gives bounds and uniqueness results.

Going further

• Taylor's theorem with remainder: Lagrange form of the error.
• Mean value theorem for integrals: Average value is attained.
• Multivariable MVT: Along line segments in ℝⁿ.