Business applications: cost, revenue and profit
Problem 3.711 · medium
A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 4 x + 200 \) dollars, and each unit sells for \( \displaystyle \$64 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.
- \[ \left. \frac{d}{d x} \left(\frac{x^{2}}{20} + 4 x + 200\right) \right|_{\substack{ x=20 }} = 6 \]Marginal cost C′(20): about the cost of unit 21.✓ Proved
- \[ 64 x - \left(\frac{x^{2}}{20} + 4 x + 200\right) = - \frac{x^{2}}{20} + 60 x - 200 \]Profit = revenue − cost.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{20} + 60 x - 200\right) = 60 - \frac{x}{10} \]P′(x).✓ Proved
- \[ \left. 60 - \frac{x}{10} \right|_{\substack{ x=600 }} = 0 \]P′ = 0 at x = 600.✓ Proved
- \[ \frac{d^{2}}{d x^{2}} \left(- \frac{x^{2}}{20} + 60 x - 200\right) = - \frac{1}{10} \]P″ < 0: a maximum.✓ Proved
- \[ \left. 60 x + \frac{\left(-1\right) x^{2}}{20} - 200 \right|_{\substack{ x=600 }} = 17800 \]The maximum profit.✓ Proved
Answer \( C'(20) = 6,\ x = 600,\ P(600) = 17800 \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ 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 | profit tried at every whole number of units from 0 to 20000; marginal cost by a central difference |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (misleading) — The comment on line 1 claims C'(20) is the cost of unit 21, which is a common misconception; marginal cost is the instantaneous rate of change, not the discrete cost of the next unit.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The comment on line 1 claims C'(20) is the cost of unit 21, which is a common misconception; marginal cost is the instantaneous rate of change, not the discrete cost of the next unit.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15gpt-oss:20b: pass 2026-10-15
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/marginal_analysis, checked 2026-10-18 with SymPy 1.14.0.