---
title: "State and prove Lagrange's mean value theorem"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:lagrange-mvt-state-prove"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/lagrange-mvt-state-prove
site: BCT Engineering
---

# State and prove Lagrange's mean value theorem

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2071 Shrawan, 5 marks; 2069 Chaitra; 2068 Shrawan; 2067 Asar (asked 4 times)
- Video: [Real Analysis | Mean Value Theorem | Lagrange's Mean Value Theorem - Proof & Examples](https://www.youtube.com/watch?v=_rLizW7giT4) — Dr.Gajendra Purohit and 2 more
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/lagrange-mvt-state-prove
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

State and prove Lagrange's Mean Value theorem.

## Short answer

If f is continuous on [a,b] and differentiable on (a,b), there exists c in (a,b) such that f'(c) = (f(b)−f(a))/(b−a). Proof: define φ(x) = f(x) − f(a) − k(x−a) with k = (f(b)−f(a))/(b−a); φ satisfies Rolle’s theorem, so φ'(c)=0 yields f'(c)=k.

## Answer

#### Statement
If a function f is (i) continuous on the closed interval [a, b] and (ii) differentiable on the open interval (a, b), then there is at least one point c in (a, b) such that

f′(c) = (f(b) − f(a)) / (b − a).

Geometrically: somewhere on the curve between A(a, f(a)) and B(b, f(b)) the tangent is parallel to the chord AB.

#### Proof
Rolle's theorem is used. Define

φ(x) = f(x) − f(a) − k(x − a), where k = (f(b) − f(a)) / (b − a).

1. φ is continuous on [a, b] and differentiable on (a, b), because f is, and k(x − a) is a polynomial.
2. φ(a) = f(a) − f(a) − 0 = 0.
3. φ(b) = f(b) − f(a) − k(b − a) = f(b) − f(a) − (f(b) − f(a)) = 0.

So φ(a) = φ(b) = 0. By Rolle's theorem there is c in (a, b) with φ′(c) = 0.

Now φ′(x) = f′(x) − k, so f′(c) − k = 0, that is

f′(c) = (f(b) − f(a)) / (b − a).

This proves the theorem. Writing b = a + h and c = a + θh (0 < θ < 1), it takes the form f(a + h) = f(a) + h f′(a + θh).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which conditions does Lagrange's MVT require on f over [a, b]?
   A. Continuous on [a, b], differentiable on (a, b) and f(a) = f(b)
   B. Continuous on [a, b] and differentiable on (a, b)
   C. Continuous on (a, b) and f'(x) > 0
   D. Differentiable on [a, b] only
   Answer: B. Lagrange's MVT needs only continuity on the closed interval and differentiability on the open interval; f(a) = f(b) is the extra Rolle condition. (Key point: Hypotheses)

2. In the proof, which function is defined so that Rolle's theorem can be applied?
   A. phi(x) = f'(x) - k
   B. phi(x) = f(x) + k(x - a)
   C. phi(x) = f(x) - f(b)
   D. phi(x) = f(x) - f(a) - k(x - a), k = (f(b) - f(a))/(b - a)
   Answer: D. Subtracting the chord line from f gives phi with phi(a) = phi(b) = 0. (Key point: Auxiliary function)

3. Why is phi(a) = phi(b) = 0?
   A. Because phi' is zero at both ends
   B. Because k = 0
   C. Because f(a) = f(b) is given
   D. Because phi(a) = 0 trivially and phi(b) = f(b) - f(a) - k(b - a) = 0 by the choice of k
   Answer: D. k is chosen so that k(b - a) = f(b) - f(a), making phi vanish at both ends. (Key point: Why Rolle applies)

4. What does phi'(c) = 0 give?
   A. f(c) = k
   B. f'(c) = 0
   C. f'(c) = f(b) - f(a)
   D. f'(c) = k = (f(b) - f(a))/(b - a)
   Answer: D. phi'(x) = f'(x) - k, so phi'(c) = 0 means f'(c) = k, the chord slope. (Key point: Conclusion)

## Close variants set in other exams

- [Prove Lagrange's MVT; verify for log x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Alagrange-mvt-verify-log-x) — Mathematics I, chapter 1, asked 5×
- [Prove Lagrange's MVT using Rolle's theorem](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Arolle-to-lagrange) — Mathematics I, chapter 1, asked 5×

## Related in Mathematics I

- [Rolle vs Lagrange; verify Lagrange's MVT](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Arolle-lagrange-difference-verify) — Mathematics I, chapter 1

## Related in other subjects

- [State and prove final value theorem](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afinal-value-theorem-prove) — Applied Mathematics, chapter 2, asked 5×
- [Final value theorem of z-transform](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Afinal-value-theorem) — Digital Control System, chapter 2
- [State and prove Green's theorem](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Astate-and-prove-greens-theorem) — Mathematics III, chapter 2, asked 11×
- [State and prove uniqueness theorem](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Auniqueness-theorem-state-prove) — Electromagnetism, chapter 2, asked 4×