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Optimization

Problem 3.167 · medium

Two nonnegative numbers add up to 24. What is the largest possible value of their product?

If one number is x, the other is 24 − x, and the product is P(x) = x(24 − x) for 0 ≤ x ≤ 24.
Reviewed
\[ \frac{d}{d x} x \left(24 - x\right) = 24 - 2 x \]
P'(x).✓ Proved
\[ \left. 24 - 2 x \right|_{\substack{ x=12 }} = 0 \]
P'(x) = 0 at x = 12.✓ Proved
\[ \left. x \left(24 - x\right) \right|_{\substack{ x=12 }} = 144 \]
At the endpoints P = 0, so this is the maximum.✓ Proved

Answer \( 144 \)

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 a maximum by comparing with the endpoints. The logic is sound and complete.

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 a maximum by comparing with the endpoints. The logic is sound and complete.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (misleading) 2026-09-27 — Line 4 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). While true in this specific case, relying solely on endpoint comparison without confirming the nature of the critical point is logically incomplete and teaches a flawed method for general optimization problems.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly sets up the optimization problem, identifies the critical point, and verifies it is a maximum by comparing with endpoints.
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.