---
title: "Minimum of x²+xy+y²+3z² under constraint"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:minimum-x2-xy-y2-3z2"
source: https://www.bctengineer.com/q/semester_2/mathematics/minimum-x2-xy-y2-3z2
site: BCT Engineering
---

# Minimum of x²+xy+y²+3z² under constraint

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2072 Ashwin, 5 marks; 2068 Bhadra, 5 marks (asked 2 times)
- Video: [Lagrange Multipliers | Geometric Meaning & Full Example](https://www.youtube.com/watch?v=8mjcnxGMwFo) — Dr. Trefor Bazett
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/minimum-x2-xy-y2-3z2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

Find the minimum value of the function x²+xy+y²+3z² under the condition x+2y+4z = 60.

## Short answer

Solve ∇f = k∇g for f = x²+xy+y²+3z² and constraint x+2y+4z=60. Stationary point: x=0, y=90/7, z=60/7. The positive definite form ensures a minimum: 2700/7.

## Answer

Minimise f = x² + xy + y² + 3z² subject to g = x + 2y + 4z − 60 = 0.

#### Lagrange conditions
Let F = f + λ(x + 2y + 4z − 60). Then (taking −λ as the multiplier k, so ∇f = k∇g)

```text
2x + y  = k          ... (1)
x + 2y  = 2k         ... (2)
6z      = 4k         ... (3)
x + 2y + 4z = 60     ... (4)
```

#### Solving
- (2) − 2·(1): x + 2y − 4x − 2y = 2k − 2k, so −3x = 0, hence x = 0.
- Then (1) gives y = k, and (2) gives 2y = 2k, which is consistent.
- From (3), z = 2k/3.
- Put these in (4): 0 + 2k + 8k/3 = 60, so 14k/3 = 60, k = 90/7.

So x = 0, y = 90/7, z = 60/7.

#### Value
```text
f = y² + 3z² = (90/7)² + 3(60/7)²
  = 8100/49 + 10800/49
  = 18900/49 = 2700/7 ≈ 385.71
```

#### Why it is a minimum
The quadratic form x² + xy + y² + 3z² has a matrix with eigenvalues 1/2, 3/2 and 3 (all positive), so f is positive definite and convex. A convex function restricted to a plane has its minimum at its stationary point.

**Answer: minimum value = 2700/7 ≈ 385.71, at (0, 90/7, 60/7).**

## Check yourself

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

1. With k as multiplier (∇f = k∇g), which equation comes from the z-derivative?
   A. 2z = 4k
   B. 6z = 4k
   C. 6z = k
   D. 3z = 4k
   Answer: B. ∂f/∂z = 6z and ∂g/∂z = 4, so 6z = 4k. (Key point: Multiplier equations)

2. Subtracting 2×(eq 1) from (eq 2) gives what?
   A. x = 0
   B. x = y
   C. z = 0
   D. y = 0
   Answer: A. x+2y-2(2x+y) = -3x = 0, so x = 0. (Key point: Solving)

3. What is the stationary point?
   A. (0, 30, 0)
   B. (90/7, 0, 60/7)
   C. (0, 90/7, 60/7)
   D. (0, 60/7, 90/7)
   Answer: C. k = 90/7, y = k = 90/7, z = 2k/3 = 60/7, x = 0. (Key point: Stationary point)

4. Why is the stationary value a minimum?
   A. Because the constraint is linear
   B. Because x = 0
   C. The quadratic form is positive definite (all eigenvalues positive), so f is convex
   D. Because k is positive
   Answer: C. Eigenvalues 1/2, 3/2, 3 are positive, so f is convex and its stationary point on the plane is the minimum (2700/7). (Key point: Why minimum)

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