The Mean Value Theorem and Rolle's theorem
Problem 3.600 · medium
Verify that \( \displaystyle f(x) = - 3 x^{2} - 4 x + 2 \) 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.
- \[ - \frac{\left. - 3 x^{2} - 4 x + 2 \right|_{\substack{ x=-2 }}}{3} + \frac{\left. - 3 x^{2} - 4 x + 2 \right|_{\substack{ x=1 }}}{3} = -1 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(- 3 x^{2} - 4 x + 2\right) = - 6 x - 4 \]Differentiate.✓ Proved
- \[ -1 \]c = -1/2 solves f′(c) = -1 and lies in (-2, 1).✓ 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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly solve the equation f'(c) = -1 to find c. It merely states that c = -1/2 is a solution without showing the algebraic step (-6c - 4 = -1 => c = -1/2), which is the core of the problem. Additionally, the notation in step 2 is obscure and potentially misleading regarding the calculation of the secant slope.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly solve the equation f'(c) = -1 to find c. It merely states that c = -1/2 is a solution without showing the algebraic step (-6c - 4 = -1 => c = -1/2), which is the core of the problem. Additionally, the notation in step 2 is obscure and potentially misleading regarding the calculation of the secant slope.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The calculation of the secant slope in step 2 is incorrect; the numerator is f(1) - f(-2) = -5 - (-10) = 5, and the denominator is 1 - (-2) = 3, so the slope is 5/3, not -1. Consequently, the value c = -1/2 is incorrect.
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 2026-10-12 with SymPy 1.14.0.