The Mean Value Theorem and Rolle's theorem
Problem 3.601 · medium
Verify that \( \displaystyle f(x) = 2 x^{3} + 2 x - 1 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [1, 5] \), 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
- \[ - \frac{\left. 2 x^{3} + 2 x - 1 \right|_{\substack{ x=1 }}}{4} + \frac{\left. 2 x^{3} + 2 x - 1 \right|_{\substack{ x=5 }}}{4} = 64 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(2 x^{3} + 2 x - 1\right) = 6 x^{2} + 2 \]Differentiate.✓ Proved
- \[ 64 \]c = sqrt(93)/3 solves f′(c) = 64 and lies in (1, 5).✓ Proved
Answer \( c = \frac{\sqrt{93}}{3} \)
Lines: 3 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 |
| 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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly calculate f(1) and f(5) or show the secant slope calculation, jumping directly to the result 64. More critically, it asserts that c = sqrt(93)/3 solves f'(c) = 64 without showing the algebraic steps (solving 6c^2 + 2 = 64), and it does not verify that this c lies within the interval (1, 5), which is required by the theorem.
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 2026-10-12 with SymPy 1.14.0.