The Mean Value Theorem and Rolle's theorem
Problem 3.498 · medium
Verify that \( \displaystyle f(x) = 2 x^{3} - 12 x^{2} + 16 x \) 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.
- \[ - \frac{\left. 2 x^{3} - 12 x^{2} + 16 x \right|_{\substack{ x=0 }}}{2} + \frac{\left. 2 x^{3} - 12 x^{2} + 16 x \right|_{\substack{ x=2 }}}{2} = 0 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(2 x^{3} - 12 x^{2} + 16 x\right) = 6 x^{2} - 24 x + 16 \]Differentiate.✓ Proved
- \[ -32 + 6 \left(2 - \frac{2 \sqrt{3}}{3}\right)^{2} + 16 \sqrt{3} = 0 \]c = 2 - 2*sqrt(3)/3 solves f′(c) = 0 and lies in (0, 2).✓ Proved
Answer \( c = 2 - \frac{2 \sqrt{3}}{3} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to find all numbers c guaranteed by the theorem. The quadratic equation f'(x) = 0 has two roots in (0, 2), but only one is listed. Additionally, Step 2 calculates the average rate of change as 0, but does not explicitly state that f(0) = f(2), which is the specific hypothesis of Rolle's Theorem required.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to find all numbers c guaranteed by the theorem. The quadratic equation f'(x) = 0 has two roots in (0, 2), but only one is listed. Additionally, Step 2 calculates the average rate of change as 0, but does not explicitly state that f(0) = f(2), which is the specific hypothesis of Rolle's Theorem required.gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to verify the condition f(a) = f(b) explicitly, instead calculating the slope of the secant line (which is 0, but this is not the standard hypothesis check). More critically, it only provides one value for c, whereas the quadratic equation f'(x)=0 has two roots in (0, 2), and the problem asks for 'every number c'.
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.