Business applications: cost, revenue and profit
Problem 3.706 · medium
A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 4 x + 100 \) dollars, and each unit sells for \( \displaystyle \$14 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.
- \[ \left. \frac{d}{d x} \left(\frac{x^{2}}{50} + 4 x + 100\right) \right|_{\substack{ x=10 }} = \frac{22}{5} \]Marginal cost C′(10): about the cost of unit 11.✓ Proved
- \[ 14 x - \left(\frac{x^{2}}{50} + 4 x + 100\right) = - \frac{x^{2}}{50} + 10 x - 100 \]Profit = revenue − cost.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{50} + 10 x - 100\right) = 10 - \frac{x}{25} \]P′(x).✓ Proved
- \[ \left. 10 - \frac{x}{25} \right|_{\substack{ x=250 }} = 0 \]P′ = 0 at x = 250.✓ Proved
- \[ \frac{d^{2}}{d x^{2}} \left(- \frac{x^{2}}{50} + 10 x - 100\right) = - \frac{1}{25} \]P″ < 0: a maximum.✓ Proved
- \[ \left. 10 x + \frac{\left(-1\right) x^{2}}{50} - 100 \right|_{\substack{ x=250 }} = 1150 \]The maximum profit.✓ Proved
Answer \( C'(10) = \frac{22}{5},\ x = 250,\ P(250) = 1150 \)
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: pass — The solution correctly derives the marginal cost, profit function, and maximizes profit using standard calculus techniques. All steps are logically sound and algebraically verified.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly derives the marginal cost, profit function, and maximizes profit using standard calculus techniques. All steps are logically sound and algebraically verified.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (misleading) 2026-10-16 — The comment in line 1 claiming that marginal cost is 'about the cost of unit 11' is incorrect; C'(10) approximates the cost of the 11th unit, but the phrasing implies it is the cost of unit 11 itself, which is a common misconception. More importantly, the problem asks for the number of units that maximizes profit, and while x=250 is correct, the solution does not explicitly state that x must be non-negative or integer, though 250 is an integer. The primary issue is the misleading definition of marginal cost in the comment.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly derives the profit function, identifies the critical point, verifies it is a maximum using the second derivative test, and calculates the correct values for marginal cost and maximum profit.gpt-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.