The Mean Value Theorem and Rolle's theorem
Problem 3.499 · medium
Verify that \( \displaystyle f(x) = 3 x^{2} - x + 2 \) 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.
- \[ - \frac{\left. 3 x^{2} - x + 2 \right|_{\substack{ x=1 }}}{2} + \frac{\left. 3 x^{2} - x + 2 \right|_{\substack{ x=3 }}}{2} = 11 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(3 x^{2} - x + 2\right) = 6 x - 1 \]Differentiate.✓ Proved
- \[ 11 \]c = 2 solves f′(c) = 11 and lies in (1, 3).✓ Proved
Answer \( c = 2 \)
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) = 11 to find c. It merely states that c=2 is a solution without showing the derivation (6c - 1 = 11 => 6c = 12 => c = 2), which is the core computational step required by the problem.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly solve the equation f'(c) = 11 to find c. It merely states that c=2 is a solution without showing the derivation (6c - 1 = 11 => 6c = 12 => c = 2), which is the core computational step required by the problem.gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — Line 2 calculates the average of the function values at the endpoints, not the slope of the secant line (which requires the difference quotient). Although the numerical result 11 happens to be correct, the formula stated is wrong and would teach a student an incorrect method.
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 2026-10-08 with SymPy 1.14.0.