Lagrange multipliers
Problem 10.202 · medium
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
- Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.Reviewed
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(3 x + 3 y\right)\\\frac{\partial}{\partial y} \left(3 x + 3 y\right)\end{matrix}\right] = \left[\begin{matrix}3\\3\end{matrix}\right] \]∇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 = 3/(2λ), y = 3/(2λ); substituting into the constraint gives λ = ±√(18)/(2√1).Reviewed
- \[ 3 \sqrt{2} \]The maximum; the minimum is its negative.✓ Proved
Answer \( \max = 3 \sqrt{2},\ \min = - 3 \sqrt{2} \)
Lines: 3 proved, 2 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly sets up the Lagrange multiplier equations, solves for the critical points, and identifies the correct maximum and minimum values. The logic is sound and the final answer matches the stated answer.
Every verdict on record (4)
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 correct maximum and minimum values. The logic is sound and the final answer matches the stated answer.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly solve for x and y, jumping directly to the final values without showing the substitution steps clearly. More critically, line 4 contains a calculation error in the derivation of lambda (it should be lambda = +/- 3/sqrt(2) or similar, not +/- sqrt(18)/2 which is actually correct but the notation is messy) and line 5 lists the maximum twice instead of listing the maximum and minimum values or points. The final answer provided in the problem statement is just the maximum, but the problem asks for both max and min.gpt-oss:20b: fail (misleading) 2026-09-27 — The solution incorrectly states that both the maximum and minimum values are 3√2; the minimum should be –3√2.
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 2026-09-27 with SymPy 1.14.0.