Lagrange multipliers
Problem 10.259 · medium Video
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 4 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
- Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(4 x + 4 y\right)\\\frac{\partial}{\partial y} \left(4 x + 4 y\right)\end{matrix}\right] = \left[\begin{matrix}4\\4\end{matrix}\right] \]∇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 = 4/(2λ); substituting into the constraint gives λ = ±√(32)/(2√25).
- \[ 20 \sqrt{2} \]The maximum; the minimum is its negative.
Answer \( \max = 20 \sqrt{2},\ \min = - 20 \sqrt{2} \)
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: passqwen3.6:27b-mlx: fail (error) — The derivation of lambda in step 4 is algebraically incorrect (it claims lambda = +/- sqrt(32)/(2*5) but the correct value is +/- 2/sqrt(5)). Furthermore, the solution fails to explicitly calculate the maximum and minimum values of f, jumping directly to the final answer without showing the substitution of the critical points into f.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The derivation of lambda in step 4 is algebraically incorrect (it claims lambda = +/- sqrt(32)/(2*5) but the correct value is +/- 2/sqrt(5)). Furthermore, the solution fails to explicitly calculate the maximum and minimum values of f, jumping directly to the final answer without showing the substitution of the critical points into f.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to explicitly state or calculate the minimum value, which is -20*sqrt(2). While it mentions the minimum is the negative of the maximum, the final answer line only lists the maximum value twice, omitting the required minimum.gpt-oss:20b: pass 2026-09-29 — The solution correctly sets up the Lagrange multiplier equations and solves for λ, x, and y. The only issue is a minor style error: the final line repeats the maximum value twice instead of listing the minimum as its negative. No mathematical mistakes are present.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to explicitly state the minimum value, which is required by the problem statement. Additionally, the derivation of lambda in step 4 is algebraically messy and skips the intermediate step of solving for x and y, making the logic hard to follow.gpt-oss:20b: pass 2026-09-29
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.