The Mean Value Theorem and Rolle's theorem
Problem 3.768 · medium
Verify that \( \displaystyle f(x) = - x^{3} + 3 x + 2 \) satisfies the hypotheses of Rolle's 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.
- \[ \frac{\left(-1\right) \left. - x^{3} + 3 x + 2 \right|_{\substack{ x=-1 }}}{3} + \frac{\left. - x^{3} + 3 x + 2 \right|_{\substack{ x=2 }}}{3} = 0 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(- x^{3} + 3 x + 2\right) = 3 - 3 x^{2} \]Differentiate.✓ Proved
- \[ \left. 3 - 3 x^{2} \right|_{\substack{ x=1 }} = 0 \]c = 1 solves f′(c) = 0 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. How we verify
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(2), which is a required hypothesis for Rolle's Theorem. Instead, it computes the slope of the secant line, which is relevant for the Mean Value Theorem but does not establish the equality of endpoint values needed for Rolle's Theorem.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-01 — The solution fails to verify that f(-1) = f(2), which is a required hypothesis for Rolle's Theorem. Instead, it computes the slope of the secant line, which is relevant for the Mean Value Theorem but does not establish the equality of endpoint values needed for Rolle's Theorem.gpt-oss:20b: pass 2026-11-01qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution fails to verify the crucial hypothesis that f(-1) = f(2). Line 2 calculates the slope of the secant line, but does not explicitly state or prove that the function values at the endpoints are equal, which is required for Rolle's Theorem. Without this, the hypotheses are not fully verified.gpt-oss:20b: pass 2026-10-31qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution fails to verify that f(-1) = f(2), which is a required hypothesis for Rolle's Theorem. Line 2 calculates the average rate of change (slope of secant) rather than checking the equality of function values, and the arithmetic in Line 2 is incorrect for that purpose as well.gpt-oss:20b: pass 2026-10-31
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 2026-11-01 with SymPy 1.14.0.