The Mean Value Theorem and Rolle's theorem
Problem 3.495 · medium
Verify that \( \displaystyle f(x) = 2 x^{3} - x^{2} + 2 x + 2 \) 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
- \[ - \frac{\left. 2 x^{3} - x^{2} + 2 x + 2 \right|_{\substack{ x=-2 }}}{4} + \frac{\left. 2 x^{3} - x^{2} + 2 x + 2 \right|_{\substack{ x=2 }}}{4} = 10 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(2 x^{3} - x^{2} + 2 x + 2\right) = 6 x^{2} - 2 x + 2 \]Differentiate.✓ Proved
- \[ 10 \]c = -1 solves f′(c) = 10 and lies in (-2, 2).✓ Proved
- \[ 10 \]c = 4/3 solves f′(c) = 10 and lies in (-2, 2).✓ Proved
Answer \( c = -1,\ \frac{4}{3} \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 identifies that polynomials satisfy the MVT hypotheses, calculates the secant slope correctly, and verifies that the stated values of c satisfy f'(c) = slope within the interval.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies that polynomials satisfy the MVT hypotheses, calculates the secant slope correctly, and verifies that the stated values of c satisfy f'(c) = slope within the interval.gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies that polynomial functions satisfy the hypotheses of the Mean Value Theorem on any closed interval. The calculations for the secant slope, the derivative, and the resulting values of c are correct and lie within the specified interval.
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.