The Mean Value Theorem and Rolle's theorem
Problem 3.501 · medium
Verify that \( \displaystyle f(x) = x^{3} - x \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-1, 2] \), 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.
- \[ - \frac{\left. x^{3} - x \right|_{\substack{ x=-1 }}}{3} + \frac{\left. x^{3} - x \right|_{\substack{ x=2 }}}{3} = 2 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(x^{3} - x\right) = 3 x^{2} - 1 \]Differentiate.✓ Proved
- \[ 2 \]c = 1 solves f′(c) = 2 and lies in (-1, 2).✓ Proved
Answer \( c = 1 \)
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 solve the equation f'(c) = 2 for c, instead merely verifying that c=1 is a solution. It omits the other valid solution c = -1, which also lies in the interval (-1, 2) and satisfies the Mean Value Theorem.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to solve the equation f'(c) = 2 for c, instead merely verifying that c=1 is a solution. It omits the other valid solution c = -1, which also lies in the interval (-1, 2) and satisfies the Mean Value Theorem.gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution claims c=1 is the only solution, but c=-1 is also a solution to f'(c)=2 and lies in the interval (-1, 2). The problem asks for 'every number c', so omitting c=-1 is an error.
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.