---
title: "Prove Lagrange's MVT; verify for log x"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:lagrange-mvt-verify-log-x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/lagrange-mvt-verify-log-x
site: BCT Engineering
---

# Prove Lagrange's MVT; verify for log x

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2069 Asar
- 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-verify-log-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

State and prove Lagrange's mean value theorem and verify f(x) = log x, x ∈ [1, e].

## Short answer

Lagrange's MVT: If f is continuous on [a,b] and differentiable on (a,b), ∃c∈(a,b) with f'(c) = (f(b)−f(a))/(b−a). Proof: define φ(x)=f(x)−f(a)−k(x−a), k=(f(b)−f(a))/(b−a); φ(a)=φ(b)=0; by Rolle's, φ'(c)=0 ⇒ f'(c)=k. Verify: f(x)=log x on [1,e], f(1)=0, f(e)=1, slope 1/(e−1); f'(x)=1/x, so c=e−1 ≈1.718 in (1,e).

## Answer

#### Statement
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), where k = (f(b) − f(a))/(b − a).

- φ is continuous on [a, b] and differentiable on (a, b).
- φ(a) = 0 and φ(b) = f(b) − f(a) − k(b − a) = 0.

By Rolle's theorem there is c in (a, b) with φ′(c) = 0. But φ′(x) = f′(x) − k, so f′(c) = k = (f(b) − f(a))/(b − a). ∎

#### Verification for f(x) = log x on [1, e]
1. log x is continuous on [1, e] and differentiable on (1, e), since x > 0.
2. f(1) = 0, f(e) = 1, so the chord slope is (1 − 0)/(e − 1) = 1/(e − 1).
3. f′(x) = 1/x, so f′(c) = 1/c = 1/(e − 1), giving c = e − 1 ≈ 1.718.

Since 1 < 1.718 < e ≈ 2.718, c lies in (1, e).

**Lagrange's MVT is verified with c = e − 1.**

## Check yourself

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

1. The proof uses phi(x) =
   A. f(x) - f(a) - k(x-a)
   B. f(x) - k
   C. f'(x) - k
   D. f(x) + f(a)
   Answer: A. It vanishes at a and b. (Key point: Auxiliary function)

2. For log x on [1, e], the chord slope is:
   A. e - 1
   B. 1
   C. 1/e
   D. 1/(e - 1)
   Answer: D. (1 - 0)/(e - 1). (Key point: Chord slope)

3. 1/c = 1/(e-1) gives:
   A. c = 1/(e-1)
   B. c = 1
   C. c = e
   D. c = e - 1
   Answer: D. Invert both sides. (Key point: Value of c)

## 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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