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Lagrange multipliers

Problem 10.56 · medium Video

Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 2 y \) on the circle \( \displaystyle x^2 + y^2 = 9 \).

Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
Reviewed
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(4 x + 2 y\right)\\\frac{\partial}{\partial y} \left(4 x + 2 y\right)\end{matrix}\right] = \left[\begin{matrix}4\\2\end{matrix}\right] \]
∇f.✓ Proved
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 9\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 9\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]
∇g.✓ Proved
So x = 4/(2λ), y = 2/(2λ); substituting into the constraint gives λ = ±√(20)/(2√9).
Reviewed
\[ 6 \sqrt{5} \]
The maximum; the minimum is its negative.Reviewed

Answer \( \max = 6 \sqrt{5},\ \min = - 6 \sqrt{5} \)

A contour map of f(x, y) = 4x + 2y: each curve joins points where f has the same value. The constraint circle of radius 3 is drawn over it; the largest and smallest values on the circle are where a level line just touches it.

Lines: 2 proved, 3 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. How we verify

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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — 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.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — 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 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly identifies the maximum value as 6*sqrt(5) (approx 13.42), whereas the correct maximum is 9*sqrt(5) (approx 20.12). The error stems from step 4, where the substitution into the constraint is performed incorrectly, leading to a wrong value for lambda and subsequently the wrong extrema.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — 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 2026-09-26

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.