The Mean Value Theorem and Rolle's theorem
Problem 3.403 · medium
Verify that \( \displaystyle f(x) = - x^{3} + x^{2} - x - 1 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [0, 3] \), 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.
- \[ - \frac{\left. - x^{3} + x^{2} - x - 1 \right|_{\substack{ x=0 }}}{3} + \frac{\left. - x^{3} + x^{2} - x - 1 \right|_{\substack{ x=3 }}}{3} = -7 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(- x^{3} + x^{2} - x - 1\right) = - 3 x^{2} + 2 x - 1 \]Differentiate.✓ Proved
- \[ - 3 \left(\frac{1}{3} + \frac{\sqrt{19}}{3}\right)^{2} - \frac{1}{3} + \frac{2 \sqrt{19}}{3} = -7 \]c = 1/3 + sqrt(19)/3 solves f′(c) = -7 and lies in (0, 3).✓ Proved
Answer \( c = \frac{1}{3} + \frac{\sqrt{19}}{3} \)
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 (style) — [domain objection, downgraded to style] The solution fails to verify that the found value of c lies within the interval (0, 3). Since c = 1/3 + sqrt(19)/3 is approximately 1.91, it is in the interval, but the text claims it solves f'(c) = -7 while the equation shown is -3*c^2 + 2*c - 1 = -7, which is correct, but the solution ignores the other root of the quadratic equation, c = 1/3 - sqrt(19)/3, which is negative and thus outside the interval. While the final answer is correct, the reasoning is incomplete because it does not explicitly check the domain for both roots or explain why the other root is rejected, potentially misleading a student into thinking there is only one solution to the quadratic without checking bounds.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The solution fails to verify that the found value of c lies within the interval (0, 3). Since c = 1/3 + sqrt(19)/3 is approximately 1.91, it is in the interval, but the text claims it solves f'(c) = -7 while the equation shown is -3*c^2 + 2*c - 1 = -7, which is correct, but the solution ignores the other root of the quadratic equation, c = 1/3 - sqrt(19)/3, which is negative and thus outside the interval. While the final answer is correct, the reasoning is incomplete because it does not explicitly check the domain for both roots or explain why the other root is rejected, potentially misleading a student into thinking there is only one solution to the quadratic without checking bounds.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to verify that the found value c lies within the open interval (0, 3). Since c = (1 + sqrt(19))/3 is approximately 1.9, it is in the interval, but the text claims it solves f'(c) = -7 while the equation shown is incorrect (it omits the constant term -1 from the derivative and has sign errors). Furthermore, the quadratic equation -3c^2 + 2c - 1 = -7 yields two solutions, but only one is in (0, 3); the solution ignores the other root and does not explicitly check the interval constraint for the valid root.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.