The Mean Value Theorem and Rolle's theorem
Problem 3.404 · medium
Verify that \( \displaystyle f(x) = x^{3} - 3 x \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-2, 1] \), 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.
- \[ - \frac{\left. x^{3} - 3 x \right|_{\substack{ x=-2 }}}{3} + \frac{\left. x^{3} - 3 x \right|_{\substack{ x=1 }}}{3} = 0 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(x^{3} - 3 x\right) = 3 x^{2} - 3 \]Differentiate.✓ Proved
- \[ 0 \]c = -1 solves f′(c) = 0 and lies in (-2, 1).✓ 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 explicitly solve the equation f'(c) = 0 to find all values of c. It jumps to checking c = -1 without showing that c = 1 is also a solution to 3c^2 - 3 = 0, and crucially, it does not verify that c = 1 is excluded because it is not in the open interval (-2, 1). This omission teaches the student that finding one root is sufficient, rather than finding all roots and filtering by the interval.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly solve the equation f'(c) = 0 to find all values of c. It jumps to checking c = -1 without showing that c = 1 is also a solution to 3c^2 - 3 = 0, and crucially, it does not verify that c = 1 is excluded because it is not in the open interval (-2, 1). This omission teaches the student that finding one root is sufficient, rather than finding all roots and filtering by the interval.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly verify that the found value c = -1 lies within the open interval (-2, 1), which is a required part of the Mean Value Theorem conclusion. Additionally, the intermediate steps for calculating the secant slope and solving the derivative equation are presented as unchecked equations without showing the algebraic work, making it impossible to verify the correctness of the arithmetic or the logic connecting the steps.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.