The Statement

If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one point c in (a, b) where:

f'(c) = [f(b) − f(a)] / (b − a)

Somewhere between a and b, the instantaneous slope equals the average slope of the line joining the endpoints.

Driving Analogy

If you drove 120 miles in 2 hours, your average speed was 60 mph. The MVT guarantees that at some moment during the trip, your speedometer read exactly 60 mph.

Geometric Meaning

Draw the chord from (a, f(a)) to (b, f(b)). The MVT guarantees that somewhere on the curve, the tangent line is parallel to that chord. Visually clear, but the proof requires the intermediate value theorem and careful argument.

Rolle's Theorem

The special case where f(a) = f(b): there exists c where f'(c) = 0. Every smooth bump has a flat top.

Worked Example

📋 Verify MVT for f(x) = x² on [1, 3]
Average rate[f(3)−f(1)]/(3−1) = (9−1)/2 = 4
Set f'(c) = 42c = 4 → c = 2
Checkc = 2 ∈ (1,3) ✓

Frequently Asked Questions

Can there be more than one c?▾
Yes — the MVT guarantees "at least one." A wavy function might have many points where the tangent is parallel to the secant.
What is the MVT used for?▾
It proves that positive derivative implies increasing function, zero derivative everywhere implies constant function, and it underlies the proof of L'Hôpital's rule and Taylor's theorem.

Formal Statement and Conditions

The MVT requires two conditions: f must be continuous on the closed interval [a, b], and differentiable on the open interval (a, b). Both are necessary. The classic counterexample is f(x) = |x| on [−1, 1]: it is continuous but not differentiable at 0, and the MVT fails — the average rate of change is 0 but f'(x) is never 0.

Rolle's Theorem — The Special Case

Rolle's Theorem is the MVT when f(a) = f(b). The conclusion simplifies: there exists c where f'(c) = 0. Geometrically: if a curve starts and ends at the same height, it must have a horizontal tangent somewhere in between.

Rolle's is used to prove the MVT: apply it to the auxiliary function g(x) = f(x) − L(x) where L(x) is the secant line. Since g(a) = g(b) = 0, Rolle's gives a point where g'(c) = 0, which translates directly into the MVT conclusion.

What the MVT Proves

The MVT is primarily a theoretical tool. The results it proves are used constantly in calculus, even if you never cite the MVT explicitly:

📋 Key consequences of the MVT
IncreasingIf f'(x) > 0 on (a,b), then f is strictly increasing on [a,b]. Proof: apply MVT to any two points.
ConstantIf f'(x) = 0 everywhere, then f is constant. This justifies the "+C" in antiderivatives.
Antiderivative uniquenessAny two antiderivatives of the same function differ by a constant — a direct consequence.
Error boundsLinear approximation error: |f(x) − f(a) − f'(a)(x−a)| ≤ M|x−a|²/2 where M bounds |f''|.

Cauchy's Mean Value Theorem

A generalisation: if f and g are both continuous on [a,b] and differentiable on (a,b), then there exists c where [f(b)−f(a)]·g'(c) = [g(b)−g(a)]·f'(c). The ordinary MVT is the special case g(x) = x. Cauchy's version is used to prove L'Hôpital's Rule.

Numerical Application — Bounding Errors

If you know |f'(x)| ≤ M on [a, b], the MVT tells you the function cannot change by more than M·(b−a) over the interval. This gives error bounds in numerical analysis: if you compute f at some point, how far off can you be if the input has a small error?

📋 Error bound example
ProblemEstimate how much sin(x) changes if x changes by at most 0.01
MVT bound|sin(x+h) − sin(x)| = |cos(c)|·|h| ≤ 1·0.01 = at most 0.01
AM
Dr. Aisha Malik, PhD Mathematics
Senior Lecturer in Applied Mathematics · 12 years teaching calculus
Dr. Malik holds a PhD in Applied Mathematics from the University of Edinburgh and has taught calculus to over 4,000 students. She has reviewed this article for mathematical accuracy and pedagogical clarity.
Technically reviewed by: Prof. James Chen, Stanford Mathematics Department · April 2026