The Mean Value Theorem and Rolle's theorem
Problem 3.548 · medium
Verify that \( \displaystyle f(x) = x^{3} - 2 x \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [1, 4] \), 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.
- \[ - \frac{\left. x^{3} - 2 x \right|_{\substack{ x=1 }}}{3} + \frac{\left. x^{3} - 2 x \right|_{\substack{ x=4 }}}{3} = 19 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(x^{3} - 2 x\right) = 3 x^{2} - 2 \]Differentiate.✓ Proved
- \[ 19 \]c = sqrt(7) solves f′(c) = 19 and lies in (1, 4).✓ Proved
Answer \( c = \sqrt{7} \)
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 actually solve the equation f'(c) = 19 for c. It merely states that c = sqrt(7) is a solution without showing the algebraic steps (3c^2 - 2 = 19 => c^2 = 7 => c = sqrt(7)) or verifying that -sqrt(7) is excluded because it is not in (1, 4).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to actually solve the equation f'(c) = 19 for c. It merely states that c = sqrt(7) is a solution without showing the algebraic steps (3c^2 - 2 = 19 => c^2 = 7 => c = sqrt(7)) or verifying that -sqrt(7) is excluded because it is not in (1, 4).gpt-oss:20b: pass 2026-10-10gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The calculation of the secant slope in step 2 is incorrect; the formula used is invalid and yields 19 instead of the correct value 13. Consequently, the derived value c = sqrt(7) 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 2026-10-10 with SymPy 1.14.0.