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The Mean Value Theorem and Rolle's theorem

Problem 3.398 · medium

Verify that \( \displaystyle f(x) = - x^{3} - 2 x^{2} - x - 2 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-1, 1] \), 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.
\[ - \frac{\left. - x^{3} - 2 x^{2} - x - 2 \right|_{\substack{ x=-1 }}}{2} + \frac{\left. - x^{3} - 2 x^{2} - x - 2 \right|_{\substack{ x=1 }}}{2} = -2 \]
The slope of the secant line.✓ Proved
\[ \frac{d}{d x} \left(- x^{3} - 2 x^{2} - x - 2\right) = - 3 x^{2} - 4 x - 1 \]
Differentiate.✓ Proved
\[ - \frac{4 \sqrt{7}}{3} - 3 \left(- \frac{2}{3} + \frac{\sqrt{7}}{3}\right)^{2} + \frac{5}{3} = -2 \]
c = -2/3 + sqrt(7)/3 solves f′(c) = -2 and lies in (-1, 1).✓ Proved

Answer \( c = - \frac{2}{3} + \frac{\sqrt{7}}{3} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to find all numbers c guaranteed by the Mean Value Theorem. The equation f'(c) = -2 yields two solutions, c = (-2 ± sqrt(7))/3, but only one is listed. Additionally, the text in line 4 incorrectly claims that the single value found is the only solution, omitting the second valid root c = (-2 - sqrt(7))/3 which also lies in (-1, 1).

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to find all numbers c guaranteed by the Mean Value Theorem. The equation f'(c) = -2 yields two solutions, c = (-2 ± sqrt(7))/3, but only one is listed. Additionally, the text in line 4 incorrectly claims that the single value found is the only solution, omitting the second valid root c = (-2 - sqrt(7))/3 which also lies in (-1, 1).
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to find all numbers c guaranteed by the Mean Value Theorem. The equation f'(c) = -2 yields two solutions, c = (-2 + sqrt(7))/3 and c = (-2 - sqrt(7))/3, both of which lie in (-1, 1). The solution only lists one.
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.