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Lagrange multipliers

Problem 10.120 · medium Video

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

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
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 16\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 16\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]
∇g.✓ Proved
So x = 1/(2λ), y = 2/(2λ); substituting into the constraint gives λ = ±√(5)/(2√16).
\[ 4 \sqrt{5} \]
The maximum; the minimum is its negative.

Answer \( \max = 4 \sqrt{5},\ \min = - 4 \sqrt{5} \)

A contour map of f(x, y) = x + 2y: 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 fails to explicitly state the minimum value, which is required by the problem statement. Additionally, the derivation in step 4 is presented as a single leap without showing the intermediate algebraic substitution, making it difficult to verify the logic.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly state the minimum value, which is required by the problem statement. Additionally, the derivation in step 4 is presented as a single leap without showing the intermediate algebraic substitution, making it difficult to verify the logic.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The derivation of lambda in step 4 is algebraically incorrect (it yields lambda = +/- 2/sqrt(5) instead of +/- sqrt(5)/4). Furthermore, step 5 incorrectly claims the maximum and minimum are both 4*sqrt(5), whereas the minimum should be -4*sqrt(5).
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The derivation in step 4 is algebraically incorrect; solving x = 1/(2λ) and y = 1/λ leads to λ = ±1/(4√5), not ±√5/(2√16). Although the final numerical answer is correct, the intermediate logic is flawed.
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.