Optimization
Problem 3.85 · medium
Two nonnegative numbers add up to 20. What is the largest possible value of their product?
- If one number is x, the other is 20 − x, and the product is P(x) = x(20 − x) for 0 ≤ x ≤ 20.
- \[ \frac{d}{d x} x \left(20 - x\right) = 20 - 2 x \]P'(x).✓ Proved
- \[ \left. 20 - 2 x \right|_{\substack{ x=10 }} = 0 \]P'(x) = 0 at x = 10.✓ Proved
- \[ \left. x \left(20 - x\right) \right|_{\substack{ x=10 }} = 100 \]At the endpoints P = 0, so this is the maximum.✓ Proved
Answer \( 100 \)
Lines: 3 proved, 1 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 | ✓ 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 | the objective sampled at 200,001 points of its interval tops out at the same value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution claims the maximum is found because the endpoints are 0, but it fails to verify that the critical point is a maximum (e.g., via the second derivative test or checking the sign of the first derivative). While the conclusion is correct, the reasoning is logically incomplete and does not justify why the critical point is a maximum rather than a minimum.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution claims the maximum is found because the endpoints are 0, but it fails to verify that the critical point is a maximum (e.g., via the second derivative test or checking the sign of the first derivative). While the conclusion is correct, the reasoning is logically incomplete and does not justify why the critical point is a maximum rather than a minimum.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (misleading) 2026-09-27 — The solution claims the result is the maximum because the endpoints are 0, but it fails to verify that the critical point is a local maximum (e.g., via the second derivative test or first derivative sign change). Without this check, the critical point could theoretically be a minimum or inflection point.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution claims the maximum is 100 because the endpoints are 0, but it fails to verify that the critical point is a maximum (e.g., via the second derivative test or checking values). While the conclusion is correct, the reasoning is logically incomplete and potentially misleading for a student who might assume any critical point with zero endpoints is a maximum.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/optimization, checked 2026-09-28 with SymPy 1.14.0.