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The Mean Value Theorem and Rolle's theorem

Problem 3.769 · medium

Verify that \( \displaystyle f(x) = - 2 x^{3} + 10 x^{2} + 8 x - 40 \) satisfies the hypotheses of Rolle's 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.
\[ \frac{\left(-1\right) \left. - 2 x^{3} + 10 x^{2} + 8 x - 40 \right|_{\substack{ x=-2 }}}{4} + \frac{\left. - 2 x^{3} + 10 x^{2} + 8 x - 40 \right|_{\substack{ x=2 }}}{4} = 0 \]
The slope of the secant line.✓ Proved
\[ \frac{d}{d x} \left(- 2 x^{3} + 10 x^{2} + 8 x - 40\right) = - 6 x^{2} + 20 x + 8 \]
Differentiate.✓ Proved
\[ \left. - 6 x^{2} + 20 x + 8 \right|_{\substack{ x=- \frac{\sqrt{37}}{3} + \frac{5}{3} }} = 0 \]
c = 5/3 - sqrt(37)/3 solves f′(c) = 0 and lies in (-2, 2).✓ Proved

Answer \( c = \frac{5}{3} - \frac{\sqrt{37}}{3} \)

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. How we verify

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 verify the hypothesis f(a) = f(b), instead calculating the slope of the secant line. Furthermore, it omits the second valid value for c, 5/3 + sqrt(37)/3, which also lies in the interval (-2, 2).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-01 — The solution fails to verify the hypothesis f(a) = f(b), instead calculating the slope of the secant line. Furthermore, it omits the second valid value for c, 5/3 + sqrt(37)/3, which also lies in the interval (-2, 2).
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution fails to find all numbers c guaranteed by the theorem. The quadratic derivative has two roots, but only one is listed. Additionally, the check for the second root's location in the interval is missing.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution fails to verify that f(-2) = f(2), which is a required hypothesis for Rolle's Theorem (f(-2)=-16 and f(2)=16, so they are not equal). Additionally, it only provides one of the two values of c guaranteed by the theorem.
gpt-oss:20b: pass 2026-10-31

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 2026-11-01 with SymPy 1.14.0.