Lagrange multipliers
Problem 10.201 · medium
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
- Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x + y\right)\\\frac{\partial}{\partial y} \left(x + y\right)\end{matrix}\right] = \left[\begin{matrix}1\\1\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 4\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 4\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]∇g.✓ Proved
- So x = 1/(2λ), y = 1/(2λ); substituting into the constraint gives λ = ±√(2)/(2√4).
- \[ 2 \sqrt{2} \]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 (error) — The solution misapplies the Lagrange multiplier equations: it incorrectly solves for λ and consequently gives wrong values for x, y, and λ. The final statement also repeats the maximum value twice and omits the correct minimum value.qwen3.6:27b-mlx: fail (error) — The derivation of lambda in step 4 is algebraically incorrect (it yields lambda = +/- 1/sqrt(2), not +/- sqrt(2)/4). Furthermore, step 5 incorrectly claims the minimum is the negative of the maximum value listed, whereas the maximum and minimum values are 2*sqrt(2) and -2*sqrt(2) respectively.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The derivation of lambda in step 4 is algebraically incorrect (it yields lambda = +/- 1/sqrt(2), not +/- sqrt(2)/4). Furthermore, step 5 incorrectly claims the minimum is the negative of the maximum value listed, whereas the maximum and minimum values are 2*sqrt(2) and -2*sqrt(2) respectively.gpt-oss:20b: fail (error) 2026-09-27 — The solution misapplies the Lagrange multiplier equations: it incorrectly solves for λ and consequently gives wrong values for x, y, and λ. The final statement also repeats the maximum value twice and omits the correct minimum value.qwen3.6:27b-mlx: fail (error) 2026-09-27 — The derivation of lambda in step 4 is algebraically incorrect (it yields lambda = +/- 1/sqrt(2), not +/- sqrt(2)/4). Furthermore, step 5 incorrectly claims the maximum and minimum are both 2*sqrt(2); the minimum is -2*sqrt(2).gpt-oss:20b: pass 2026-09-27
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.