The Mean Value Theorem and Rolle's theorem
Problem 3.450 · medium
Verify that \( \displaystyle f(x) = x^{2} - 3 x - 4 \) satisfies the hypotheses of the Mean Value 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.
- \[ - \frac{\left. x^{2} - 3 x - 4 \right|_{\substack{ x=-2 }}}{4} + \frac{\left. x^{2} - 3 x - 4 \right|_{\substack{ x=2 }}}{4} = -3 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(x^{2} - 3 x - 4\right) = 2 x - 3 \]Differentiate.✓ Proved
- \[ -3 \]c = 0 solves f′(c) = -3 and lies in (-2, 2).✓ Proved
Answer \( c = 0 \)
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 for c. It states that c=0 is a solution but does not show the equation f'(c) = -3 leading to 2c - 3 = -3, which is the core calculation required by the problem.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly solve for c. It states that c=0 is a solution but does not show the equation f'(c) = -3 leading to 2c - 3 = -3, which is the core calculation required by the problem.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The calculation of the secant slope in step 2 is incorrect; the formula used yields -3, but the actual slope (f(2)-f(-2))/(2-(-2)) is -6. Consequently, the derived value c=0 is incorrect (the correct value is c=-1.5).
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 2026-10-06 with SymPy 1.14.0.