The Mean Value Theorem and Rolle's theorem
Problem 3.449 · medium
Verify that \( \displaystyle f(x) = x^{3} + x^{2} - x - 1 \) satisfies the hypotheses of Rolle's theorem on \( \displaystyle [-1, 1] \), 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.
- \[ - \frac{\left. x^{3} + x^{2} - x - 1 \right|_{\substack{ x=-1 }}}{2} + \frac{\left. x^{3} + x^{2} - x - 1 \right|_{\substack{ x=1 }}}{2} = 0 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(x^{3} + x^{2} - x - 1\right) = 3 x^{2} + 2 x - 1 \]Differentiate.✓ Proved
- \[ 0 \]c = 1/3 solves f′(c) = 0 and lies in (-1, 1).✓ Proved
Answer \( c = \frac{1}{3} \)
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(1), which is a required hypothesis for Rolle's Theorem. Additionally, it omits the root c = -1, which is also a valid solution to f'(c) = 0 in the interval (-1, 1).
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to verify that f(-1) = f(1), which is a required hypothesis for Rolle's Theorem. Additionally, it omits the root c = -1, which is also a valid solution to f'(c) = 0 in the interval (-1, 1).gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to verify that f(-1) = f(1), which is a required hypothesis for Rolle's Theorem. Additionally, it omits finding the second valid value c = -1, providing an incomplete answer.
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.