Optimization
Problem 3.76 · medium
Two nonnegative numbers add up to 16. What is the largest possible value of their product?
- If one number is x, the other is 16 − x, and the product is P(x) = x(16 − x) for 0 ≤ x ≤ 16.
- \[ \frac{d}{d x} x \left(16 - x\right) = 16 - 2 x \]P'(x).✓ Proved
- \[ \left. 16 - 2 x \right|_{\substack{ x=8 }} = 0 \]P'(x) = 0 at x = 8.✓ Proved
- \[ \left. x \left(16 - x\right) \right|_{\substack{ x=8 }} = 64 \]At the endpoints P = 0, so this is the maximum.✓ Proved
Answer \( 64 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
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 |
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 2026-09-26 with SymPy 1.14.0.