The Mean Value Theorem and Rolle's theorem
Problem 3.599 · medium
Verify that \( \displaystyle f(x) = - 2 x^{3} + 16 x^{2} - 38 x + 24 \) satisfies the hypotheses of Rolle's theorem on \( \displaystyle [1, 3] \), 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.
- \[ - \frac{\left. - 2 x^{3} + 16 x^{2} - 38 x + 24 \right|_{\substack{ x=1 }}}{2} + \frac{\left. - 2 x^{3} + 16 x^{2} - 38 x + 24 \right|_{\substack{ x=3 }}}{2} = 0 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(- 2 x^{3} + 16 x^{2} - 38 x + 24\right) = - 6 x^{2} + 32 x - 38 \]Differentiate.✓ Proved
- \[ - \frac{32 \sqrt{7}}{3} - 6 \left(\frac{8}{3} - \frac{\sqrt{7}}{3}\right)^{2} + \frac{142}{3} = 0 \]c = 8/3 - sqrt(7)/3 solves f′(c) = 0 and lies in (1, 3).✓ Proved
Answer \( c = \frac{8}{3} - \frac{\sqrt{7}}{3} \)
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 verify that f(1) = f(3), which is a required hypothesis for Rolle's Theorem. Additionally, it only identifies one of the two values of c guaranteed by the theorem (the other being 8/3 + sqrt(7)/3), failing the instruction to find 'every' number.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to verify that f(1) = f(3), which is a required hypothesis for Rolle's Theorem. Additionally, it only identifies one of the two values of c guaranteed by the theorem (the other being 8/3 + sqrt(7)/3), failing the instruction to find 'every' number.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to verify that f(1) = f(3), which is a required hypothesis for Rolle's Theorem. Additionally, it only identifies one of the two values of c guaranteed by the theorem (the other is 8/3 + sqrt(7)/3), contradicting the instruction to find 'every number c'.
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 2026-10-12 with SymPy 1.14.0.