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Lagrange multipliers

Problem 10.117 · medium Video

Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 4 \).

Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(4 x + 3 y\right)\\\frac{\partial}{\partial y} \left(4 x + 3 y\right)\end{matrix}\right] = \left[\begin{matrix}4\\3\end{matrix}\right] \]
∇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 = 4/(2λ), y = 3/(2λ); substituting into the constraint gives λ = ±√(25)/(2√4).
\[ 10 \]
The maximum; the minimum is its negative.

Answer \( \max = 10,\ \min = -10 \)

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

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 (misleading) — The solution incorrectly states that both the maximum and minimum values are 10; the minimum should be –10. The reasoning is otherwise correct, but the conclusion is false.
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 2026-09-28 — 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 (misleading) 2026-09-28 — The solution incorrectly states that both the maximum and minimum values are 10; the minimum should be –10. The reasoning is otherwise correct, but the conclusion is false.
qwen3.6:27b-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 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the method of Lagrange multipliers, derives the correct critical values, and identifies the maximum and minimum accurately.
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.