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The Mean Value Theorem and Rolle's theorem

Problem 3.546 · medium

Verify that \( \displaystyle f(x) = 2 x^{2} + 4 x - 2 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-1, 2] \), 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.
\[ - \frac{\left. 2 x^{2} + 4 x - 2 \right|_{\substack{ x=-1 }}}{3} + \frac{\left. 2 x^{2} + 4 x - 2 \right|_{\substack{ x=2 }}}{3} = 6 \]
The slope of the secant line.✓ Proved
\[ \frac{d}{d x} \left(2 x^{2} + 4 x - 2\right) = 4 x + 4 \]
Differentiate.✓ Proved
\[ 6 \]
c = 1/2 solves f′(c) = 6 and lies in (-1, 2).✓ Proved

Answer \( c = \frac{1}{2} \)

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 (misleading) — The solution fails to explicitly solve the equation f'(c) = 6 for c, instead just stating that c=1/2 is a solution. It also omits the crucial step of verifying that this c lies within the open interval (-1, 2), which is required by the Mean Value Theorem.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (misleading) 2026-10-10 — The solution fails to explicitly solve the equation f'(c) = 6 for c, instead just stating that c=1/2 is a solution. It also omits the crucial step of verifying that this c lies within the open interval (-1, 2), which is required by the Mean Value Theorem.
gpt-oss:20b: pass 2026-10-10
gpt-oss:20b: fail (error) 2026-10-10 — The average rate of change is computed incorrectly: (f(2)-f(-1))/(2-(-1)) = 20/3, not 6. Consequently the subsequent steps are based on a wrong value.
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly solve the equation f'(c) = 6 for c, jumping directly to the verification that c=1/2 is a solution. It also does not demonstrate that this is the unique solution, which is required to 'find every number c'.

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.