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Lagrange multipliers

Problem 10.67 · medium Video

Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 3 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).

Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(3 x + 4 y\right)\\\frac{\partial}{\partial y} \left(3 x + 4 y\right)\end{matrix}\right] = \left[\begin{matrix}3\\4\end{matrix}\right] \]
∇f.✓ Proved
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 1\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 1\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]
∇g.✓ Proved
So x = 3/(2λ), y = 4/(2λ); substituting into the constraint gives λ = ±√(25)/(2√1).
\[ 5 \]
The maximum; the minimum is its negative.

Answer \( \max = 5,\ \min = -5 \)

A contour map of f(x, y) = 3x + 4y: each curve joins points where f has the same value. The constraint circle of radius 1 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: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution incorrectly defines the constraint function as g(x, y) = x² + y² − r² = 0 with an undefined variable r, whereas the problem specifies the unit circle (r=1). Furthermore, the final answer '5' is incomplete as it fails to state the minimum value (-5) explicitly, despite the problem asking for both.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-09-28 — [domain objection, downgraded to style] The solution incorrectly defines the constraint function as g(x, y) = x² + y² − r² = 0 with an undefined variable r, whereas the problem specifies the unit circle (r=1). Furthermore, the final answer '5' is incomplete as it fails to state the minimum value (-5) explicitly, despite the problem asking for both.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly defines g(x, y) with a generic 'r' in the sentence but uses r=1 in the equations, creating a contradiction. Furthermore, the final answer '5' is incomplete as the problem asks for both maximum and minimum values.
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 points, and identifies the maximum and minimum values.
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.