The Mean Value Theorem and Rolle's theorem
Problem 3.497 · medium
Verify that \( \displaystyle f(x) = 2 x^{3} - 24 x^{2} + 82 x - 60 \) satisfies the hypotheses of Rolle's theorem on \( \displaystyle [1, 5] \), 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.
- \[ - \frac{\left. 2 x^{3} - 24 x^{2} + 82 x - 60 \right|_{\substack{ x=1 }}}{4} + \frac{\left. 2 x^{3} - 24 x^{2} + 82 x - 60 \right|_{\substack{ x=5 }}}{4} = 0 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(2 x^{3} - 24 x^{2} + 82 x - 60\right) = 6 x^{2} - 48 x + 82 \]Differentiate.✓ Proved
- \[ -110 + 6 \left(4 - \frac{\sqrt{21}}{3}\right)^{2} + 16 \sqrt{21} = 0 \]c = 4 - sqrt(21)/3 solves f′(c) = 0 and lies in (1, 5).✓ Proved
Answer \( c = 4 - \frac{\sqrt{21}}{3} \)
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: fail (error) — The solution fails to find all values of c guaranteed by Rolle's theorem. The quadratic equation f'(x) = 0 has two roots, c = 4 ± sqrt(21)/3, both of which lie in the interval (1, 5). The solution only identifies one root.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to find all values of c guaranteed by Rolle's theorem. The quadratic equation f'(x) = 0 has two roots, c = 4 ± sqrt(21)/3, both of which lie in the interval (1, 5). The solution only identifies one root.gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to verify that f(1) = f(5), which is a required hypothesis for Rolle's Theorem. Additionally, the stated answer is incomplete because it omits the second root c = 4 + sqrt(21)/3, which also lies in the interval (1, 5).
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 2026-10-08 with SymPy 1.14.0.