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Optimization

Problem 3.79 · medium

Two nonnegative numbers add up to 12. What is the largest possible value of their product?

If one number is x, the other is 12 − x, and the product is P(x) = x(12 − x) for 0 ≤ x ≤ 12.
Reviewed
\[ \frac{d}{d x} x \left(12 - x\right) = 12 - 2 x \]
P'(x).✓ Proved
\[ \left. 12 - 2 x \right|_{\substack{ x=6 }} = 0 \]
P'(x) = 0 at x = 6.✓ Proved
\[ \left. x \left(12 - x\right) \right|_{\substack{ x=6 }} = 36 \]
At the endpoints P = 0, so this is the maximum.✓ Proved

Answer \( 36 \)

Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly sets up the optimization problem, identifies the critical point, and verifies it is the maximum by comparing with endpoints.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly sets up the optimization problem, identifies the critical point, and verifies it is the maximum by comparing with endpoints.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution identifies the critical point but fails to verify it is a maximum (e.g., via the second derivative test or checking endpoints). It incorrectly claims the value at x=6 is the maximum solely because the endpoints are zero, ignoring the possibility of other critical points or local minima.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (misleading) 2026-09-26 — The sentence claims the maximum is determined because the endpoints are 0, but it fails to explicitly verify that x=6 is a maximum (e.g., via the second derivative test or sign change of P'). While the conclusion is correct, the reasoning is incomplete and could mislead a student into thinking endpoint comparison alone is sufficient without checking the nature of the critical point.
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.