The Mean Value Theorem and Rolle's theorem
Problem 3.397 · medium
Verify that \( \displaystyle f(x) = 2 x^{3} + 4 x^{2} - 22 x - 24 \) 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.
- \[ - \frac{\left. 2 x^{3} + 4 x^{2} - 22 x - 24 \right|_{\substack{ x=-1 }}}{4} + \frac{\left. 2 x^{3} + 4 x^{2} - 22 x - 24 \right|_{\substack{ x=3 }}}{4} = 0 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(2 x^{3} + 4 x^{2} - 22 x - 24\right) = 6 x^{2} + 8 x - 22 \]Differentiate.✓ Proved
- \[ - \frac{82}{3} + 6 \left(- \frac{2}{3} + \frac{\sqrt{37}}{3}\right)^{2} + \frac{8 \sqrt{37}}{3} = 0 \]c = -2/3 + sqrt(37)/3 solves f′(c) = 0 and lies in (-1, 3).✓ Proved
Answer \( c = - \frac{2}{3} + \frac{\sqrt{37}}{3} \)
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 the hypothesis f(-1) = f(3), instead calculating the slope of the secant line (which is 0, but this is not the hypothesis check). Furthermore, it only provides one of the two values of c guaranteed by Rolle's Theorem (the other is c = -2/3 - sqrt(37)/3, which also lies in (-1, 3)).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to verify the hypothesis f(-1) = f(3), instead calculating the slope of the secant line (which is 0, but this is not the hypothesis check). Furthermore, it only provides one of the two values of c guaranteed by Rolle's Theorem (the other is c = -2/3 - sqrt(37)/3, which also lies in (-1, 3)).gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to verify the hypothesis f(-1) = f(3), which is required for Rolle's Theorem. Additionally, it only provides one of the two values of c guaranteed by the theorem (the other root of f'(x)=0 is also in the interval).gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.