Lagrange multipliers
Problem 10.119 · medium Video
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).
- Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x + 2 y\right)\\\frac{\partial}{\partial y} \left(x + 2 y\right)\end{matrix}\right] = \left[\begin{matrix}1\\2\end{matrix}\right] \]∇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 = 2/(2λ); substituting into the constraint gives λ = ±√(5)/(2√4).
- \[ 2 \sqrt{5} \]The maximum; the minimum is its negative.
Answer \( \max = 2 \sqrt{5},\ \min = - 2 \sqrt{5} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly solves the system of equations. From 1 = 2λx and 2 = 2λy, one gets x = 1/(2λ) and y = 1/λ, not y = 2/(2λ) (which implies y=x, leading to the wrong constraint substitution). The correct substitution yields λ = ±1/√10, giving extrema ±√10, not ±2√5.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly solves the system of equations. From 1 = 2λx and 2 = 2λy, one gets x = 1/(2λ) and y = 1/λ, not y = 2/(2λ) (which implies y=x, leading to the wrong constraint substitution). The correct substitution yields λ = ±1/√10, giving extrema ±√10, not ±2√5.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — 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 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 contains a significant algebraic error in solving for lambda; it claims lambda = ±sqrt(5)/(2*sqrt(4)) but the correct value is ±sqrt(5)/4. While the final answer is correct, the intermediate derivation is mathematically false.gpt-oss:20b: fail (error) 2026-09-26 — The step solving ∇f=λ∇g is incorrect: from 1=2λx and 2=2λy we get x=1/(2λ) and y=1/λ, not y=2/(2λ). Consequently the subsequent λ value and the claimed extrema are wrong.
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.