---
title: "Extreme value with two linear constraints"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:extreme-value-two-linear-constraints"
source: https://www.bctengineer.com/q/semester_2/mathematics/extreme-value-two-linear-constraints
site: BCT Engineering
---

# Extreme value with two linear constraints

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2070 Magh, 5 marks
- Video: [🟡15b - Lagrange's Multipliers: Two Constraints - Find the maximum and minimum | Ex 4 & 5](https://www.youtube.com/watch?v=t-4KtKBopqE) — SkanCity Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/extreme-value-two-linear-constraints
- 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+z = 1 and 2y+z = 2.

## Short answer

Use Lagrange multipliers with two constraints. Solve for x, y, z from partial derivatives and constraints, obtaining x=1/3, y=2/3, z=2/3. The function is squared distance from origin to a line, so only a minimum exists. Minimum value is 1.

## Answer

Find the extreme value of f = x² + y² + z² subject to g₁ = x + z − 1 = 0 and g₂ = 2y + z − 2 = 0.

#### Lagrange conditions
F = x² + y² + z² + λ(x + z − 1) + μ(2y + z − 2)

```text
∂F/∂x = 2x + λ = 0        →  x = −λ/2
∂F/∂y = 2y + 2μ = 0       →  y = −μ
∂F/∂z = 2z + λ + μ = 0    →  z = −(λ + μ)/2
```

#### Using the constraints
Let a = −λ/2 and b = −μ. Then x = a, y = b, z = a + b/2.

- x + z = 1: a + a + b/2 = 1, so 2a + b/2 = 1.
- 2y + z = 2: 2b + a + b/2 = 2, so a + 5b/2 = 2.

Solve: from the first, b = 2 − 4a. Put into the second: a + 5 − 10a = 2, so a = 1/3, b = 2/3.

So x = 1/3, y = 2/3, z = 1/3 + 1/3 = 2/3.

Check: x + z = 1 ✓.  2y + z = 4/3 + 2/3 = 2 ✓.

#### Value
f = 1/9 + 4/9 + 4/9 = 1.

#### Nature
The two constraints define a straight line, and f is the squared distance from the origin to a point of the line. A line has a nearest point but no farthest, so the value is a minimum.

**Answer: minimum value = 1, at (1/3, 2/3, 2/3).**

## Check yourself

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

1. How many multipliers are used for the two constraints x+z = 1 and 2y+z = 2?
   A. None
   B. Two, λ and μ
   C. One
   D. Three
   Answer: B. One per constraint. (Key point: Multipliers)

2. The stationary point is:
   A. (1/3, 1/3, 2/3)
   B. (1/3, 2/3, 2/3)
   C. (2/3, 1/3, 1/3)
   D. (1, 0, 0)
   Answer: B. a = 1/3, b = 2/3 give x = 1/3, y = 2/3, z = 2/3; both constraints check. (Key point: Solution)

3. The value 1 is a minimum because:
   A. x, y, z are positive
   B. λ is positive
   C. f is linear
   D. The constraints define a line and f is squared distance from the origin, which has a nearest but no farthest point
   Answer: D. A line has a closest point to the origin but no farthest. (Key point: Value and nature)

## Close variants set in other exams

- [Extreme value of x²+y²+z² with ax+by+cz=p](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aextreme-value-x2-y2-z2) — Mathematics, chapter 1, asked 5×
- [Extreme value of x²+y²+z² with x+y+z=3a](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aextreme-value-x2-y2-z2-2) — Mathematics, chapter 1, asked 5×

## Related in Mathematics

- [Extreme values with two constraints](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aextreme-values-two-constraints) — Mathematics, chapter 1, asked 2×
- [Extreme value of x²+y²+z² with x+y+z=1](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aextreme-value-x2-y2-z2-3) — Mathematics, chapter 1
- [Initial value problem y''+2y'+5y=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ainitial-value-problem-y-2y) — Mathematics, chapter 4, asked 3×

## Related in other subjects

- [Short note: Two magic tees connected in H-plane](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Atwo-magic-tees-h-plane-note) — RF and Microwave Engineering, chapter 4
- [Two blocks connected over a pulley](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Ablocks-pulley-friction-acceleration) — Engineering Mechanics, chapter 2
- [Two magic tees connected at E-arms](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Atwo-magic-tees-e-arms) — RF and Microwave Engineering, chapter 4
- [Linear solution techniques; CG two iterations](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Asolution-techniques-cg-numerical-two-iter) — Computational Techniques in Civil Engineering, chapter 2