Lagrange multipliers
Problem 10.233 · medium
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
- Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x + 4 y\right)\\\frac{\partial}{\partial y} \left(x + 4 y\right)\end{matrix}\right] = \left[\begin{matrix}1\\4\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 25\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 25\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]∇g.✓ Proved
- So x = 1/(2λ), y = 4/(2λ); substituting into the constraint gives λ = ±√(17)/(2√25).
- \[ 5 \sqrt{17} \]The maximum; the minimum is its negative.✓ Proved
Answer \( \max = 5 \sqrt{17},\ \min = - 5 \sqrt{17} \)
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 computed value of f at the critical points is ±45/√17, not ±5√17. The algebra in step 4 leads to λ=±√17/10, and substituting back gives f=9/(2λ)=±45/√17, so the stated maximum/minimum are incorrect.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 4 contains a significant algebraic error in solving for lambda. Substituting x = 1/(2λ) and y = 4/(2λ) into x² + y² = 25 yields 17/(4λ²) = 25,
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 4 contains a significant algebraic error in solving for lambda. Substituting x = 1/(2λ) and y = 4/(2λ) into x² + y² = 25 yields 17/(4λ²) = 25,gpt-oss:20b: fail (error) 2026-09-28 — The computed value of f at the critical points is ±45/√17, not ±5√17. The algebra in step 4 leads to λ=±√17/10, and substituting back gives f=9/(2λ)=±45/√17, so the stated maximum/minimum are incorrect.qwen3.6:27b-mlx: pass 2026-09-28 — 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 2026-09-28
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 2026-09-28 with SymPy 1.14.0.