---
title: "Extreme value of x²+y²+z² with x+y+z=3a"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:extreme-value-x2-y2-z2-2"
source: https://www.bctengineer.com/q/semester_2/mathematics/extreme-value-x2-y2-z2-2
site: BCT Engineering
---

# Extreme value of x²+y²+z² with x+y+z=3a

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2072 Magh, 5 marks
- Video: [DifferentialCalculus_2 | 85. Find the minimum value of x² + y² + z² when x + y + z = 3a](https://www.youtube.com/watch?v=2HdtCw4wlvk) — Bhuvaneshwari N EE & EC
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/extreme-value-x2-y2-z2-2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

Find the extreme value of x²+y²+z² connected by the relation x+y+z = 3a.

## Short answer

Using Lagrange multiplier, solve ∇(x²+y²+z²) = λ∇(x+y+z−3a) to get x=y=z=a. The extreme value is 3a². Since f is squared distance on a plane, this is a minimum.

## Answer

Find the extreme value of f = x² + y² + z² subject to x + y + z = 3a using Lagrange's multiplier.

#### Setting up
F = x² + y² + z² + λ(x + y + z − 3a)

```text
2x + λ = 0
2y + λ = 0
2z + λ = 0
```

Thus x = y = z = −λ/2. The constraint gives 3x = 3a, so x = y = z = a and λ = −2a.

#### Value
f = a² + a² + a² = 3a².

#### Nature
f is the squared distance of a point on the plane x + y + z = 3a from the origin. A plane has a nearest point but no farthest point, so 3a² is a minimum. (Check: by Cauchy–Schwarz, (x + y + z)² ≤ 3(x² + y² + z²), giving f ≥ 3a².)

**Answer: extreme (minimum) value = 3a², at (a, a, a).**

## Check yourself

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

1. Extremising x²+y²+z² with x+y+z = 3a gives:
   A. (2a, a, 0)
   B. (3a, 0, 0)
   C. (a, a, a)
   D. (a/3, a/3, a/3)
   Answer: C. x = y = z = -λ/2 and 3x = 3a. (Key point: Stationary point)

2. The value there is:
   A. a²
   B. 3a²
   C. 3a
   D. 9a²
   Answer: B. 3a². (Key point: Value)

3. Why is it a minimum rather than maximum?
   A. Because λ = −2a
   B. Because a > 0
   C. Because the function is linear
   D. The plane has a nearest point to the origin but no farthest point
   Answer: D. f is the squared distance to the origin; on a plane it has a minimum only. (Key point: Nature)

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