The Mean Value Theorem and Rolle's theorem
Problem 3.766 · medium
Verify that \( \displaystyle f(x) = - 2 x^{2} - 2 x - 4 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-2, 0] \), 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
- \[ \frac{\left(-1\right) \left. - 2 x^{2} - 2 x - 4 \right|_{\substack{ x=-2 }}}{2} + \frac{\left. - 2 x^{2} - 2 x - 4 \right|_{\substack{ x=0 }}}{2} = 2 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(- 2 x^{2} - 2 x - 4\right) = - 4 x - 2 \]Differentiate.✓ Proved
- \[ \left. - 4 x - 2 \right|_{\substack{ x=-1 }} = 2 \]c = -1 solves f′(c) = 2 and lies in (-2, 0).✓ Proved
Answer \( c = -1 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
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 |
| 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 a polynomial and accurately solves for c using the Mean Value Theorem formula.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly verifies the hypotheses for a polynomial and accurately solves for c using the Mean Value Theorem formula.gpt-oss:20b: pass 2026-11-01qwen3.6:27b-mlx: pass 2026-10-31gpt-oss:20b: pass 2026-10-31qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly identifies that polynomials satisfy the MVT hypotheses, calculates the secant slope correctly, and solves for c within the interval.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.