The Mean Value Theorem and Rolle's theorem
Problem 3.762 · medium
Verify that \( \displaystyle f(x) = x^{3} - 2 x \) satisfies the hypotheses of the Mean Value 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.
- \[ \frac{\left(-1\right) \left. x^{3} - 2 x \right|_{\substack{ x=0 }}}{2} + \frac{\left. x^{3} - 2 x \right|_{\substack{ x=2 }}}{2} = 2 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(x^{3} - 2 x\right) = 3 x^{2} - 2 \]Differentiate.✓ Proved
- \[ \left. 3 x^{2} - 2 \right|_{\substack{ x=\frac{2 \sqrt{3}}{3} }} = 2 \]c = 2*sqrt(3)/3 solves f′(c) = 2 and lies in (0, 2).✓ Proved
Answer \( c = \frac{2 \sqrt{3}}{3} \)
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. How we verify
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 values of c guaranteed by the Mean Value Theorem. Solving 3c^2 - 2 = 2 yields c = ±2√3/3, but only the positive root is listed. The negative root is not in (0, 2), but the solution does not explicitly check or discard it, nor does it state that it is the unique solution in the interval.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-01 — The solution fails to find all values of c guaranteed by the Mean Value Theorem. Solving 3c^2 - 2 = 2 yields c = ±2√3/3, but only the positive root is listed. The negative root is not in (0, 2), but the solution does not explicitly check or discard it, nor does it state that it is the unique solution in the interval.gpt-oss:20b: pass 2026-11-01qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution fails to find all values of c. The equation 3c^2 - 2 = 2 yields c = ±2√3/3, but only the positive root is listed. The negative root is not in (0, 2), so it is correctly excluded, but the solution does not explicitly show the solving step or address the negative root, making the derivation incomplete and potentially misleading about how to find 'every' number.gpt-oss:20b: pass 2026-10-31qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution fails to find all values of c guaranteed by the Mean Value Theorem. The equation 3c^2 - 2 = 2 yields c = ±2√3/3, but only the positive root is listed, ignoring the negative root which is also a solution to f'(c) = 2 (though it lies outside the interval, the step 'find every number' implies checking both roots derived from the quadratic, or explicitly stating why the negative one is rejected. More critically, the prompt asks to find *every* number c. The negative root -2√3/3 is not in (0,2), so it is correctly excluded, but the solution does not explicitly show the solving of 3c^2=4 or mention the negative root, making the derivation incomplete. However, looking closer, the prompt asks to find every c *the theorem guarantees*. The theorem guarantees c in (0,2). -2√3/3 is not in (0,2). So only one c exists. The solution just states the answer. Is it an error? The step 4 says 'c = ... solves ... and lies in (0,2)'. It doesn't show the work to find it. But more importantly, did it miss any? 3c^2 - 2 = 2 => 3c^2 = 4 => c^2 = 4/3 => c = ±2/√3. Only 2/√3 is in (0,2). So the answer is correct. Why fail? Let's re-read carefully. 'Find every number c'. The solution just asserts the answer. It doesn't show the algebraic step of setting f'(c) = slope. Step 4 is an equation check, not a derivation. The sentence in step 1 is fine. Step 2 is an equation. Step 3 is an equation. Step 4 is an equation. Where is the sentence that sets up f'(c) = slope? It's missing. The solution jumps from calculating the slope and derivative to checking the final answer. It fails to explicitly state the equation f'(c) = 2 and solve for c. This is a 'style' or 'misleading' issue? Or 'error'? The instructions say 'Check that... each conclusion follows from the lines before it'. Step 4 does not follow from Step 3 without the intermediate step of equating them and solving. A student might not know how to get from 3x^2-2 to 2√3/3. It's a missing logical step. Is it an error? It's a gap in reasoning. I will call it 'misleading' because it hides the crucial step of solving the equation, or 'error' because the logical flow is broken. Let's look at the options. 'error' is for wrong steps. 'misleading' is for teaching false things. 'style' is for clarity. This is a missing step. Usually, missing steps are 'style' unless they are critical. Here, solving 3c^2-2=2 is the core of the problem. Skipping it is bad. But is it 'wrong'? No. Is it 'misleading'? It might lead a student to think you just guess c. I'll go with 'style' or 'misleading'. Actually, looking at previous examples, missing the negative root check is often an error if not mentioned. But here the negative root is outside. The solution doesn't mention it. It just gives the positive one. It's technically correct but incomplete in explanation. However, the prompt asks to 'find every number'. If I just write 'c = 2√3/3', I have found it. The verification is in step 4. The derivation is missing. I will mark it as 'style' because the answer is correct and the verification is there, but the derivation is skipped.gpt-oss:20b: pass 2026-10-31
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 2026-11-01 with SymPy 1.14.0.