The Mean Value Theorem and Rolle's theorem
Problem 3.500 · hard
Verify that \( \displaystyle f(x) = x^{3} - 5 x \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-2, 2] \), 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.Reviewed
- \[ - \frac{\left. x^{3} - 5 x \right|_{\substack{ x=-2 }}}{4} + \frac{\left. x^{3} - 5 x \right|_{\substack{ x=2 }}}{4} = -1 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(x^{3} - 5 x\right) = 3 x^{2} - 5 \]Differentiate.✓ Proved
- \[ -1 \]c = -2*sqrt(3)/3 solves f′(c) = -1 and lies in (-2, 2).✓ Proved
- \[ -1 \]c = 2*sqrt(3)/3 solves f′(c) = -1 and lies in (-2, 2).✓ Proved
Answer \( c = - \frac{2 \sqrt{3}}{3},\ \frac{2 \sqrt{3}}{3} \)
Lines: 4 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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 |
| 5 | ✓ 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: pass — The solution correctly verifies the hypotheses for the Mean Value Theorem and accurately identifies the values of c that satisfy the conclusion.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly verifies the hypotheses for the Mean Value Theorem and accurately identifies the values of c that satisfy the conclusion.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 explicitly solve the equation f'(c) = -1 to derive the values of c. It merely verifies that the final answers are correct, which does not demonstrate the process of finding 'every number c' as required by the problem statement.
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.