The Mean Value Theorem and Rolle's theorem
Problem 3.451 · medium
Verify that \( \displaystyle f(x) = - 2 x^{3} + x^{2} + 2 x - 2 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [1, 4] \), and find every number \( \displaystyle c \) the theorem guarantees.
- f is a polynomial, so it is continuous on [a, b] and differentiable on (a, b): the hypotheses hold.
- \[ - \frac{\left. - 2 x^{3} + x^{2} + 2 x - 2 \right|_{\substack{ x=1 }}}{3} + \frac{\left. - 2 x^{3} + x^{2} + 2 x - 2 \right|_{\substack{ x=4 }}}{3} = -35 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(- 2 x^{3} + x^{2} + 2 x - 2\right) = - 6 x^{2} + 2 x + 2 \]Differentiate.✓ Proved
- \[ - 6 \left(\frac{1}{6} + \frac{\sqrt{223}}{6}\right)^{2} + \frac{7}{3} + \frac{\sqrt{223}}{3} = -35 \]c = 1/6 + sqrt(223)/6 solves f′(c) = -35 and lies in (1, 4).✓ Proved
Answer \( c = \frac{1}{6} + \frac{\sqrt{223}}{6} \)
Lines: 3 proved, 1 not checked. 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.
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 | each c checked by a difference quotient; a scan of (a, b) finds no others |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution fails to verify that the found value of c actually lies within the interval (1, 4). Since sqrt(223) is approximately 14.9, c is approxim
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution fails to verify that the found value of c actually lies within the interval (1, 4). Since sqrt(223) is approximately 14.9, c is approximqwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution claims c = 1/6 + sqrt(223)/6 lies in (1, 4), but this value is approximately 2.6, which is indeed in (1, 4). However, the equation in ligpt-oss:20b: pass 2026-10-06
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/mean_value_theorem, checked 2026-10-06 with SymPy 1.14.0.