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Lagrange multipliers

Problem 10.109 · medium Video

Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 16 \).

Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x + 3 y\right)\\\frac{\partial}{\partial y} \left(x + 3 y\right)\end{matrix}\right] = \left[\begin{matrix}1\\3\end{matrix}\right] \]
∇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 = 1/(2λ), y = 3/(2λ); substituting into the constraint gives λ = ±√(10)/(2√16).
\[ 4 \sqrt{10} \]
The maximum; the minimum is its negative.

Answer \( \max = 4 \sqrt{10},\ \min = - 4 \sqrt{10} \)

A contour map of f(x, y) = x + 3y: each curve joins points where f has the same value. The constraint circle of radius 4 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 (error) — The solution incorrectly claims the minimum is the negative of the maximum. The maximum is 4*sqrt(10) and the minimum is -4*sqrt(10), but the text implies the maximum value itself is 4*sqrt(10) and the minimum is -4*sqrt(10) without explicitly stating the minimum value, or worse, it might be interpreted as the minimum being -4*sqrt(10) which is correct, but the phrasing 'the minimum is its negative' is ambiguous if not paired with the explicit value. More critically, step 4 calculates lambda but does not explicitly solve for x and y to verify the points, though the final answer is correct. However, the biggest issue is that the problem asks for BOTH maximum and minimum, but the stated answer only provides the maximum. The solution text says 'The maximum; the minimum is its negative' but the final output line only lists '4*sqrt(10), 4*sqrt(10)' which is confusing and likely incorrect formatting for a pair of answers. Actually, looking at line 5, it outputs '4*sqrt(10), 4*sqrt(10)' which suggests it thinks the max and min are the same or just repeated the max. This is a significant error in reporting the result.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly claims the minimum is the negative of the maximum. The maximum is 4*sqrt(10) and the minimum is -4*sqrt(10), but the text implies the maximum value itself is 4*sqrt(10) and the minimum is -4*sqrt(10) without explicitly stating the minimum value, or worse, it might be interpreted as the minimum being -4*sqrt(10) which is correct, but the phrasing 'the minimum is its negative' is ambiguous if not paired with the explicit value. More critically, step 4 calculates lambda but does not explicitly solve for x and y to verify the points, though the final answer is correct. However, the biggest issue is that the problem asks for BOTH maximum and minimum, but the stated answer only provides the maximum. The solution text says 'The maximum; the minimum is its negative' but the final output line only lists '4*sqrt(10), 4*sqrt(10)' which is confusing and likely incorrect formatting for a pair of answers. Actually, looking at line 5, it outputs '4*sqrt(10), 4*sqrt(10)' which suggests it thinks the max and min are the same or just repeated the max. This is a significant error in reporting the result.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly state or calculate the minimum value, which is required by the problem statement. While it mentions the minimum is the negative of the maximum, it does not provide the value -4*sqrt(10) as a final answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to state the minimum value explicitly, despite the problem asking for both maximum and minimum. While the final line mentions the minimum is the negative, the stated answer only provides the maximum.
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.