Lagrange multipliers
Problem 10.231 · medium
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).
- Solve ∇f = λ∇g with g(x, y) = x² + y² − r² = 0.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(4 x + 3 y\right)\\\frac{\partial}{\partial y} \left(4 x + 3 y\right)\end{matrix}\right] = \left[\begin{matrix}4\\3\end{matrix}\right] \]∇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 = 3/(2λ); substituting into the constraint gives λ = ±√(25)/(2√25).
- \[ 25 \]The maximum; the minimum is its negative.✓ Proved
Answer \( \max = 25,\ \min = -25 \)
Lines: 3 proved, 2 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.
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 and obscures the logic; specifically, substituting x=2/lambda and y=3/lambda into x^2+y^2=25 yields 4/lambda^2 + 9/lambda^2 = 25, so 13/lambda^2 = 25, implying lambda = +/- sqrt(13)/5, not the expression given. Furthermore, the final values for max/min are incorrect: f = 4(2/lambda) + 3(3/lambda) = 25/lambda = 25/(sqrt(13)/5) = 125/sqrt(13), not 25.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The derivation of lambda in step 4 is algebraically incorrect and obscures the logic; specifically, substituting x=2/lambda and y=3/lambda into x^2+y^2=25 yields 4/lambda^2 + 9/lambda^2 = 25, so 13/lambda^2 = 25, implying lambda = +/- sqrt(13)/5, not the expression given. Furthermore, the final values for max/min are incorrect: f = 4(2/lambda) + 3(3/lambda) = 25/lambda = 25/(sqrt(13)/5) = 125/sqrt(13), not 25.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to calculate the minimum value, which is -25, despite the problem asking for both maximum and minimum. Additionally, the derivation of lambda in step 4 is algebraically confused and skips necessary steps.gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.