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The Mean Value Theorem and Rolle's theorem

Problem 3.553 · medium

Verify that \( \displaystyle f(x) = - 2 x^{2} + 3 x + 1 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-2, 0] \), 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.
\[ - \frac{\left. - 2 x^{2} + 3 x + 1 \right|_{\substack{ x=-2 }}}{2} + \frac{\left. - 2 x^{2} + 3 x + 1 \right|_{\substack{ x=0 }}}{2} = 7 \]
The slope of the secant line.✓ Proved
\[ \frac{d}{d x} \left(- 2 x^{2} + 3 x + 1\right) = 3 - 4 x \]
Differentiate.✓ Proved
\[ 7 \]
c = -1 solves f′(c) = 7 and lies in (-2, 0).✓ 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 satisfies f'(c) = 7, but f'(-1) = 3 - 4(-1) = 7 is correct, however the secant slope calculation in step 2 is wrong. f(-2) = -11, f(0) = 1, so the slope is (1 - (-11)) / (0 - (-2)) = 12 / 2 = 6, not 7. Consequently, f'(c) should be 6, leading to 3 - 4c = 6 => c = -3/4. The stated answer c = -1 is incorrect.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution claims c = -1 satisfies f'(c) = 7, but f'(-1) = 3 - 4(-1) = 7 is correct, however the secant slope calculation in step 2 is wrong. f(-2) = -11, f(0) = 1, so the slope is (1 - (-11)) / (0 - (-2)) = 12 / 2 = 6, not 7. Consequently, f'(c) should be 6, leading to 3 - 4c = 6 => c = -3/4. The stated answer c = -1 is incorrect.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly solve the equation f'(c) = 7 for c, jumping directly to the claim that c = -1 is the solution without showing the algebraic steps. Additionally, the notation in line 2 is non-standard and confusing for calculating the average rate of change.
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.