The Mean Value Theorem and Rolle's theorem
Problem 3.452 · medium
Verify that \( \displaystyle f(x) = - 2 x^{3} + 4 x^{2} \) satisfies the hypotheses of Rolle's theorem on \( \displaystyle [0, 2] \), 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.
- \[ - \frac{\left. - 2 x^{3} + 4 x^{2} \right|_{\substack{ x=0 }}}{2} + \frac{\left. - 2 x^{3} + 4 x^{2} \right|_{\substack{ x=2 }}}{2} = 0 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(- 2 x^{3} + 4 x^{2}\right) = - 6 x^{2} + 8 x \]Differentiate.✓ Proved
- \[ 0 \]c = 4/3 solves f′(c) = 0 and lies in (0, 2).✓ Proved
Answer \( c = \frac{4}{3} \)
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: fail (misleading) — The solution skips the crucial step of showing that f(0)=f(2) and solving f′(c)=0 to find c=4/3. A student could be misled into thinking the verification is complete without these explicit checks.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly verify that f(0) = f(2), which is a required hypothesis of Rolle's Theorem. Additionally, the final step asserts that c = 4/3 is the solution without showing the algebraic derivation from f'(c) = 0.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-06 — The solution skips the crucial step of showing that f(0)=f(2) and solving f′(c)=0 to find c=4/3. A student could be misled into thinking the verification is complete without these explicit checks.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly verify that f(0) = f(2), which is a required hypothesis of Rolle's Theorem. Additionally, the final step asserts that c = 4/3 is the solution without showing the algebraic derivation from f'(c) = 0.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly verify that f(0) = f(2), which is a required hypothesis for Rolle's Theorem. Additionally, Step 2 calculates the average value of the function rather than the slope of the secant line (which should be (f(2)-f(0))/(2-0)), and Step 4 asserts the solution without showing the derivation.gpt-oss:20b: pass 2026-10-06
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.