The Mean Value Theorem and Rolle's theorem
Problem 3.402 · medium
Verify that \( \displaystyle f(x) = - 2 x^{2} + 3 x - 4 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-1, 3] \), 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.Reviewed
- \[ - \frac{\left. - 2 x^{2} + 3 x - 4 \right|_{\substack{ x=-1 }}}{4} + \frac{\left. - 2 x^{2} + 3 x - 4 \right|_{\substack{ x=3 }}}{4} = -1 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(- 2 x^{2} + 3 x - 4\right) = 3 - 4 x \]Differentiate.✓ Proved
- \[ -1 \]c = 1 solves f′(c) = -1 and lies in (-1, 3).✓ Proved
Answer \( c = 1 \)
Lines: 3 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the hypotheses for the Mean Value Theorem, calculates the average rate of change, sets up the equation f'(c) = m, and verifies the solution lies within the interval.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the hypotheses for the Mean Value Theorem, calculates the average rate of change, sets up the equation f'(c) = m, and verifies the solution lies within the interval.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The calculation of the secant slope in step 2 is incorrect; it divides by 4 instead of the interval length (3 - (-1) = 4) but applies the formula incorrectly or uses wrong values, resulting in -1 instead of the correct slope -5. Consequently, the derived value c=1 is incorrect.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.