The Mean Value Theorem and Rolle's theorem
Problem 3.549 · hard
Verify that \( \displaystyle f(x) = x^{3} - 5 x^{2} + 4 x \) satisfies the hypotheses of Rolle's theorem on \( \displaystyle [0, 4] \), 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.
- \[ - \frac{\left. x^{3} - 5 x^{2} + 4 x \right|_{\substack{ x=0 }}}{4} + \frac{\left. x^{3} - 5 x^{2} + 4 x \right|_{\substack{ x=4 }}}{4} = 0 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(x^{3} - 5 x^{2} + 4 x\right) = 3 x^{2} - 10 x + 4 \]Differentiate.✓ Proved
- \[ - \frac{38}{3} + 3 \left(\frac{5}{3} - \frac{\sqrt{13}}{3}\right)^{2} + \frac{10 \sqrt{13}}{3} = 0 \]c = 5/3 - sqrt(13)/3 solves f′(c) = 0 and lies in (0, 4).✓ Proved
- \[ - \frac{38}{3} - \frac{10 \sqrt{13}}{3} + 3 \left(\frac{\sqrt{13}}{3} + \frac{5}{3}\right)^{2} = 0 \]c = sqrt(13)/3 + 5/3 solves f′(c) = 0 and lies in (0, 4).✓ Proved
Answer \( c = \frac{5}{3} - \frac{\sqrt{13}}{3},\ \frac{\sqrt{13}}{3} + \frac{5}{3} \)
Lines: 4 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 |
| 5 | ✓ 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 verify the condition f(0) = f(4), which is required for Rolle's Theorem. Line 2 calculates the slope of the secant line (which is 0), but does not explicitly state or check that the function values at the endpoints are equal, which is the specific hypothesis of Rolle's Theorem (as opposed to the Mean Value Theorem).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to verify the condition f(0) = f(4), which is required for Rolle's Theorem. Line 2 calculates the slope of the secant line (which is 0), but does not explicitly state or check that the function values at the endpoints are equal, which is the specific hypothesis of Rolle's Theorem (as opposed to the Mean Value Theorem).gpt-oss:20b: pass 2026-10-10gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to verify the condition f(0) = f(4), which is required for Rolle's Theorem. Instead, it calculates the slope of the secant line (which is 0, but the check is not explicitly stated as f(0)=f(4)). More critically, the algebraic checks in lines 4 and 5 are incorrect; substituting the roots into the derivative 3x^2 - 10x + 4 does not yield the expressions shown (e.g., the constant term calculation is wrong), making the verification of the roots invalid.
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.