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Lagrange multipliers

Problem 10.110 · medium Video

Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).

Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(2 x + 2 y\right)\\\frac{\partial}{\partial y} \left(2 x + 2 y\right)\end{matrix}\right] = \left[\begin{matrix}2\\2\end{matrix}\right] \]
∇f.✓ Proved
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 9\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 9\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]
∇g.✓ Proved
So x = 2/(2λ), y = 2/(2λ); substituting into the constraint gives λ = ±√(8)/(2√9).
\[ 6 \sqrt{2} \]
The maximum; the minimum is its negative.

Answer \( \max = 6 \sqrt{2},\ \min = - 6 \sqrt{2} \)

A contour map of f(x, y) = 2x + 2y: each curve joins points where f has the same value. The constraint circle of radius 3 is drawn over it; the largest and smallest values on the circle are where a level line just touches it.

Lines: 2 proved, 3 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	Not checked	—	a sentence; read, not computed
5	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	f around 100,000 points of the circle tops out at the same value

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state or verify the minimum value, which is required by the problem statement. Additionally, the derivation of lambda in step 4 is presented as a single substitution step without showing the intermediate algebraic simplification, making it difficult to follow, though the final values are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly state or verify the minimum value, which is required by the problem statement. Additionally, the derivation of lambda in step 4 is presented as a single substitution step without showing the intermediate algebraic simplification, making it difficult to follow, though the final values are correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly sets up the Lagrange multiplier equations, solves for the critical points, and identifies the maximum and minimum values. The logic is sound and the final answer is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 contains a significant algebraic error in solving for lambda. From x = 1/lambda and y = 1/lambda, substituting into x^2 + y^2 = 9 gives 2/lambda^2 = 9, so lambda = +/- sqrt(2/9) = +/- sqrt(2)/3. The solution claims lambda = +/- sqrt(8)/(2*3) = +/- 2*sqrt(2)/6 = +/- sqrt(2)/3, which is actually correct numerically, but the derivation 'sqrt(8)/(2sqrt(9))' is confusingly written and implies a different path. More importantly, the final answer is correct, but the intermediate step description is messy. Wait, let me re-read carefully. x = 2/(2lambda) = 1/lambda. x^2+y^2 = 2/lambda^2 = 9 => lambda^2 = 2/9 => lambda = +/- sqrt(2)/3. The text says lambda = +/- sqrt(8)/(2sqrt(9)) = +/- (2sqrt(2))/6 = +/- sqrt(2)/3. The value is correct. Is there an error? The prompt asks to find max and min. The answer gives 6*sqrt(2). This is correct. Is there a subtle error? Step 4 says 'substituting into the constraint gives...'. It skips showing the substitution. But the result is correct. Let's look closer. Is the setup correct? Yes. Is the conclusion correct? Yes. Why would it fail? Maybe the 'sentence' in step 4 is considered 'misleading' or 'style' if it's just bad notation? Or is there a real error? Let's check the lambda calculation again. sqrt(8) = 2*sqrt(2). 2*sqrt(9) = 6. 2*sqrt(2)/6 = sqrt(2)/3. Correct. What about the max/min? f = 2x+2y. x=y=3/sqrt(2). f = 2(3/sqrt(2)) + 2(3/sqrt(2)) = 12/sqrt(2) = 6*sqrt(2). Correct. The solution seems mathematically sound, just briefly stated. However, usually 'adversarial' reviews look for missing cases or wrong logic. Did it miss the case where lambda=0? If lambda=0, grad f = 0, which is false. So no. Did it miss boundary issues? Circle is compact, f is continuous, extrema exist. Lagrange multipliers find them. The logic holds. Is there a 'style' issue? The prompt says 'Say pass with empty notes if it is sound.' It is sound. But wait, look at step 4 again. 'So x = 2/(2lambda)...'. This implies lambda != 0. This is true. Is it misleading? No. I will mark it pass.
gpt-oss:20b: fail (misleading) 2026-09-26 — The final sentence incorrectly states the maximum and minimum values as both 6√2; the minimum should be –6√2. The conclusion does not follow the previous calculations and could mislead a student.

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/lagrange_multipliers, checked 2026-10-13 with SymPy 1.14.0.