Lagrange multipliers
Problem 10.165 · medium
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 2 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
- 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
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 1\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 1\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]∇g.✓ Proved
- So x = 2/(2λ), y = 2/(2λ); substituting into the constraint gives λ = ±√(8)/(2√1).
- \[ 2 \sqrt{2} \]The maximum; the minimum is its negative.✓ Proved
Answer \( \max = 2 \sqrt{2},\ \min = - 2 \sqrt{2} \)
Lines: 3 proved, 2 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.
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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: fail (misleading) — The solution repeats the maximum value twice and omits the negative minimum value, which could mislead a student into thinking the minimum is also 2√2. The correct values are 2√2 and –2√2.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the minimum value, which is -2*sqrt(2), despite the problem asking for both maximum and minimum. Additionally, the derivation in step 4 is algebraically opaque and skips the crucial step of solving for x and y before substituting back into f(x,y).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly state the minimum value, which is -2*sqrt(2), despite the problem asking for both maximum and minimum. Additionally, the derivation in step 4 is algebraically opaque and skips the crucial step of solving for x and y before substituting back into f(x,y).gpt-oss:20b: fail (misleading) 2026-09-26 — The solution repeats the maximum value twice and omits the negative minimum value, which could mislead a student into thinking the minimum is also 2√2. The correct values are 2√2 and –2√2.qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly solves the system of equations. From 2 = 2λx and 2 = 2λy, one gets x = 1/λ and y = 1/λ, not x = 2/(2λ) (which simplifies to 1/λ, so that part is actually correct algebraically, but the substitution step is flawed). Let's re-read carefully. Step 4 says 'x = 2/(2λ), y = 2/(2λ)'. This simplifies to x=1/λ, y=1/λ. Substituting into x^2+y^2=1 gives 2/λ^2 = 1 => λ^2 = 2 => λ = ±√2. The solution claims λ = ±√(8)/(2√1) = ±2√2/2 = ±√2. So the value of λ is correct. However, the phrasing 'substituting into the constraint gives λ = ...' skips the intermediate step of finding x and y values, but the result for λ is correct. Wait, let's look at the final answer. The max value is f(x,y) = 2x+2y. If x=y=1/√2, f = 2/√2 + 2/√2 = 4/√2 = 2√2. The stated answer is 2*sqrt(2). The solution says 'The maximum; the minimum is its negative.' This is correct. Is there an error? Let's check the algebra in step 4 again. 'x = 2/(2λ)' is technically correct but weirdly written. 'y = 2/(2λ)' same. Substitution: (1/λ)^2 + (1/λ)^2 = 1 => 2/λ^2 = 1 => λ^2 = 2. The solution says λ = ±√(8)/(2√1). √8/2 = 2√2/2 = √2. So λ = ±√2. This is correct. Why would it fail? Perhaps the 'unchecked' equations are just definitions. The sentence in step 4 is the main claim. It derives λ correctly. Step 5 states the max value. It seems correct. Let me look closer at the prompt's instruction: 'find what is wrong'. Is there a subtle error? The problem asks for max and min. The solution provides max and implies min. The stated answer is just the max. The solution text says 'The maximum; the minimum is its negative.' This is a complete answer. Is the algebra in step 4 'x = 2/(2λ)' derived from 2 = λ(2x)? Yes, 2 = 2λx => x = 1/λ. 2/(2λ) is 1/λ. It's correct. Maybe the error is in the calculation of λ? λ = ±√2. The solution writes λ = ±√(8)/(2√1). This is a very strange way to write ±√2. √8/2 = √(8/4) = √2. It is correct but obfuscated. Is it misleading? No. Is it an error? No. Let's look at the constraints. g(x,y) = x^2+y^2-1. ∇g = <2x, 2y>. ∇f = <2, 2>. 2 = λ2x => x = 1/λ. 2 = λ2y => y = 1/λ. x^2+y^2=1 => 2/λ^2=1 => λ=±√2. Points: (1/√2, 1/√2) and (-1/√2, -1/√2). f(1/√2, 1/√2) = 2√2. f(-1/√2, -1/√2) = -2√2. The solution is correct. Why would I fail it? Maybe the 'unchecked' tag on the equations means I shouldn't check them, but I should check the sentences. Sentence 1: Setup. Correct. Sentence 4: Derivation. Correct result, weird intermediate form. Sentence 5: Conclusion. Correct. Wait, look at step 4 again. 'substituting into the constraint gives λ = ±√(8)/(2√1)'. The denominator is 2√1. Where did the 2 come from? If x=1/λ, x^2=1/λ^2. 2/λ^2=1. λ^2=2. λ=√2. The expression √8/2 is √2. It's correct. Is there a missing step? No. Is it possible the solution implies x and y are determined by λ, but doesn't explicitly state the points? It jumps to the value. This is standard. I will pass it unless I find a real error. Let me re-read 'x = 2/(2λ)'. This comes from 2 = λ(2x). So 2xλ = 2 => xλ = 1 => x = 1/λ. The solution writes x = 2/(2λ). This is 1/λ. It is correct. Okay, I'll pass.gpt-oss:20b: pass 2026-09-26
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 2026-09-26 with SymPy 1.14.0.