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Optimization

Problem 3.168 · medium

Two nonnegative numbers add up to 30. What is the largest possible value of their product?

If one number is x, the other is 30 − x, and the product is P(x) = x(30 − x) for 0 ≤ x ≤ 30.
\[ \frac{d}{d x} x \left(30 - x\right) = 30 - 2 x \]
P'(x).✓ Proved
\[ \left. 30 - 2 x \right|_{\substack{ x=15 }} = 0 \]
P'(x) = 0 at x = 15.✓ Proved
\[ \left. x \left(30 - x\right) \right|_{\substack{ x=15 }} = 225 \]
At the endpoints P = 0, so this is the maximum.✓ Proved

Answer \( 225 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to verify that the critical point is a maximum (e.g., via the second derivative test or analyzing the sign of the first derivative). It also incorrectly claims the endpoint values are 0 without explicitly evaluating P(0) and P(30), relying on an unproven assertion rather than calculation.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to verify that the critical point is a maximum (e.g., via the second derivative test or analyzing the sign of the first derivative). It also incorrectly claims the endpoint values are 0 without explicitly evaluating P(0) and P(30), relying on an unproven assertion rather than calculation.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (misleading) 2026-09-27 — The solution claims the result is the maximum solely because the endpoints are zero, ignoring the need to verify the critical point is a maximum (e.g., via the second derivative test or first derivative sign change). This teaches a false general principle that any critical point with lower endpoints must be a maximum.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution identifies the critical point but fails to verify it is a maximum (e.g., via the second derivative test) and incorrectly claims the endpoint values are 0 without explicitly evaluating P(0) and P(30) in the text, relying on an unproven assertion for the comparison.
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.