Lagrange multipliers
Problem 10.162 · medium
Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = x + 3 y \) on the circle \( \displaystyle x^2 + y^2 = 1 \).
- 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
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + y^{2} - 1\right)\\\frac{\partial}{\partial y} \left(x^{2} + y^{2} - 1\right)\end{matrix}\right] = \left[\begin{matrix}2 x\\2 y\end{matrix}\right] \]∇g.✓ Proved
- So x = 1/(2λ), y = 3/(2λ); substituting into the constraint gives λ = ±√(10)/(2√1).
- \[ \sqrt{10} \]The maximum; the minimum is its negative.✓ Proved
Answer \( \max = \sqrt{10},\ \min = - \sqrt{10} \)
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 solution fails to explicitly calculate the minimum value, which is -sqrt(10), despite the problem asking for both maximum and minimum. Additionally, step 4 contains a typo in the denominator (sqrt(1) instead of just 1 or r), though the final numerical result is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly calculate the minimum value, which is -sqrt(10), despite the problem asking for both maximum and minimum. Additionally, step 4 contains a typo in the denominator (sqrt(1) instead of just 1 or r), though the final numerical result is correct.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — 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 presented without showing the substitution into the constraint equation, making the logic opaque.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 2026-09-26 with SymPy 1.14.0.