---
title: "Prove Lagrange's MVT using Rolle's theorem"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:rolle-to-lagrange"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/rolle-to-lagrange
site: BCT Engineering
---

# Prove Lagrange's MVT using Rolle's theorem

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2068 Chaitra
- 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/rolle-to-lagrange
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

State Rolle's theorem and use it to prove Lagrange's mean value theorem.

## Short answer

Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b) and f(a)=f(b), then ∃c∈(a,b): f'(c)=0. To prove Lagrange's MVT, set k=(f(b)−f(a))/(b−a) and φ(x)=f(x)−f(a)−k(x−a). φ satisfies Rolle's conditions, giving φ'(c)=0 ⇒ f'(c)=k. Hence f'(c) = (f(b)−f(a))/(b−a).

## Answer

#### Rolle's theorem
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).

#### Lagrange's mean value theorem (to prove)
If f is continuous on [a, b] and differentiable on (a, b), then for some c in (a, b),

f′(c) = (f(b) − f(a))/(b − a).

#### Proof using Rolle's theorem
Let k = (f(b) − f(a))/(b − a) and define the auxiliary function

φ(x) = f(x) − f(a) − k(x − a).

1. φ is the difference of f and a linear function, so it is continuous on [a, b] and differentiable on (a, b).
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), and Rolle's theorem applies to φ: there is c in (a, b) with φ′(c) = 0.

Since φ′(x) = f′(x) − k, we get f′(c) = k, that is

**f′(c) = (f(b) − f(a))/(b − a).** ∎

If in addition f(a) = f(b), then k = 0 and the result reduces to Rolle's theorem.

## Check yourself

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

1. The auxiliary function is:
   A. phi(x) = f(x)/k
   B. phi(x) = f(x) - f(a) - k(x - a)
   C. phi(x) = f'(x) - k
   D. phi(x) = f(b) - f(x)
   Answer: B. It subtracts the chord. (Key point: Auxiliary function)

2. phi(a) = phi(b) = 0 holds since:
   A. k = 0
   B. f(a) = f(b)
   C. f' is zero
   D. k(b-a) = f(b) - f(a)
   Answer: D. k is defined as the chord slope. (Key point: Equal ends)

3. phi'(c) = 0 gives:
   A. f'(c) = k
   B. f'(c) = 0
   C. f(c) = k
   D. f''(c) = k
   Answer: A. phi' = f' - k. (Key point: Derivative)

4. If f(a) = f(b), the result reduces to:
   A. Maclaurin's series
   B. Cauchy's theorem
   C. Taylor's theorem
   D. Rolle's theorem
   Answer: D. k = 0. (Key point: Special case)

## Close variants set in other exams

- [State and prove Lagrange's mean value theorem](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Alagrange-mvt-state-prove) — Mathematics I, chapter 1, asked 6×

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