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Lagrange multipliers

Problem 10.55 · medium Video

Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).

Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
Reviewed
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(4 x + y\right)\\\frac{\partial}{\partial y} \left(4 x + y\right)\end{matrix}\right] = \left[\begin{matrix}4\\1\end{matrix}\right] \]
∇f.✓ Proved
\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 25\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 25\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]
∇g.✓ Proved
So x = 4/(2λ), y = 1/(2λ); substituting into the constraint gives λ = ±√(17)/(2√25).
Reviewed
\[ 5 \sqrt{17} \]
The maximum; the minimum is its negative.Reviewed

Answer \( \max = 5 \sqrt{17},\ \min = - 5 \sqrt{17} \)

A contour map of f(x, y) = 4x + y: each curve joins points where f has the same value. The constraint circle of radius 5 is drawn over it; the largest and smallest values on the circle are where a level line just touches it.

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 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: pass 2026-09-28
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 correct maximum and minimum values.
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 correct 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.