Home›Calculus 1›The Mean Value Theorem and Rolle's theorem›Problem 3.554

The Mean Value Theorem and Rolle's theorem

Problem 3.554 · medium

Verify that \( \displaystyle f(x) = x^{3} - 6 x \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-2, 1] \), 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.
\[ - \frac{\left. x^{3} - 6 x \right|_{\substack{ x=-2 }}}{3} + \frac{\left. x^{3} - 6 x \right|_{\substack{ x=1 }}}{3} = -3 \]
The slope of the secant line.✓ Proved
\[ \frac{d}{d x} \left(x^{3} - 6 x\right) = 3 x^{2} - 6 \]
Differentiate.✓ Proved
\[ -3 \]
c = -1 solves f′(c) = -3 and lies in (-2, 1).✓ Proved

Answer \( c = -1 \)

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 claims c = -1 is the only number guaranteed by the theorem, but c = 1 is also a solution to f'(c) = -3 and lies in the interval (-2, 1). The problem asks for 'every number', so omitting c = 1 is an error.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution claims c = -1 is the only number guaranteed by the theorem, but c = 1 is also a solution to f'(c) = -3 and lies in the interval (-2, 1). The problem asks for 'every number', so omitting c = 1 is an error.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to verify that the found value c = -1 is the only solution, as the problem asks to find 'every' number c. The quadratic equation 3c^2 - 6 = -3 has two roots, c = -1 and c = 1, but c = 1 is not in the open interval (-2, 1). The solution implicitly assumes uniqueness without showing the other root was checked and rejected.
gpt-oss:20b: pass 2026-10-10

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 2026-10-10 with SymPy 1.14.0.