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

# Minimum of x²+y²+z² with x+y+z=3a

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2074 Bhadra, 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/minimum-x2-y2-z2-x
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

Find the minimum value of x²+y²+z² when x+y+z = 3a.

## Short answer

By Lagrange multipliers or the QM-AM inequality, x=y=z=a minimizes x²+y²+z² subject to x+y+z=3a. The minimum value is 3a².

## Answer

Minimise f = x² + y² + z² subject to g = x + y + z − 3a = 0.

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

```text
∂F/∂x = 2x + λ = 0
∂F/∂y = 2y + λ = 0
∂F/∂z = 2z + λ = 0
```

So x = y = z = −λ/2. From the constraint 3x = 3a, hence

x = y = z = a   (and λ = −2a).

#### Value
f = a² + a² + a² = 3a².

#### Nature
By the Cauchy–Schwarz (or QM–AM) inequality,
(x + y + z)² ≤ 3(x² + y² + z²), so x² + y² + z² ≥ (3a)²/3 = 3a². Equality holds when x = y = z. So the stationary value is the minimum. (Geometrically, it is the squared distance from the origin to the plane x + y + z = 3a.)

**Answer: minimum value = 3a², at x = y = z = a.**

## Check yourself

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

1. Minimising x²+y²+z² subject to x+y+z = 3a gives which point?
   A. x = 3a, y = z = 0
   B. x = y = z = a/3
   C. x = y = z = a
   D. x = y = z = 3a
   Answer: C. 2x+λ = 2y+λ = 2z+λ = 0 gives x = y = z, and 3x = 3a. (Key point: Stationary point)

2. What is the minimum value?
   A. 3a²
   B. 3a
   C. a²
   D. 9a²
   Answer: A. f = a² + a² + a² = 3a². (Key point: Minimum value)

3. Which inequality confirms it is the minimum?
   A. AM ≤ GM
   B. Triangle inequality
   C. Sum of squares is always negative
   D. Cauchy–Schwarz: (x+y+z)² ≤ 3(x²+y²+z²)
   Answer: D. It gives x²+y²+z² ≥ 9a²/3 = 3a², with equality at x = y = z. (Key point: Why minimum)

## Close variants set in other exams

- [Minimum of x²+y²+z² when 1/x+1/y+1/z=1](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aminimum-x2-y2-z2-when) — Mathematics, chapter 1, asked 2×

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