Lagrange multipliers
Problem 10.262 · medium Video
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).
- Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(4 x + y\right)\\\frac{\partial}{\partial y} \left(4 x + y\right)\end{matrix}\right] = \left[\begin{matrix}4\\1\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 16\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 16\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]∇g.✓ Proved
- So x = 4/(2λ), y = 1/(2λ); substituting into the constraint gives λ = ±√(17)/(2√16).
- \[ 4 \sqrt{17} \]The maximum; the minimum is its negative.
Answer \( \max = 4 \sqrt{17},\ \min = - 4 \sqrt{17} \)
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: fail (error) — The calculation of the objective value at the critical points is incorrect. Substituting x=2/λ and y=1/(2λ) into f gives f=9/(2λ), which for λ=±√17/8 yields ±36/√17, not ±4√17. The solution misapplies the Lagrange multiplier results to obtain the wrong extrema.qwen3.6:27b-mlx: pass — The solution correctly sets up the Lagrange multiplier equations, solves for the critical points, and identifies the correct maximum and minimum values.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly sets up the Lagrange multiplier equations, solves for the critical points, and identifies the correct maximum and minimum values.gpt-oss:20b: fail (error) 2026-10-04 — The calculation of the objective value at the critical points is incorrect. Substituting x=2/λ and y=1/(2λ) into f gives f=9/(2λ), which for λ=±√17/8 yields ±36/√17, not ±4√17. The solution misapplies the Lagrange multiplier results to obtain the wrong extrema.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly sets up the Lagrange multiplier equations, solves for the critical points, and identifies the correct maximum value. The logic is sound and the final answer is correct.gpt-oss:20b: fail (error) 2026-09-29 — The solution incorrectly computes the maximum value. Using the Lagrange multiplier equations gives λ=±√17/8, leading to f=±2√17, not ±4√17. The final statement overstates the maximum and minimum by a factor of two.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly sets up the Lagrange multiplier equations, solves for x and y in terms of lambda, and correctly identifies the maximum and minimum values.gpt-oss:20b: pass 2026-09-29
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.