The Mean Value Theorem and Rolle's theorem
Problem 3.764 · medium
Verify that \( \displaystyle f(x) = x^{3} - 2 x \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [-2, 0] \), 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.
- \[ \frac{\left(-1\right) \left. x^{3} - 2 x \right|_{\substack{ x=-2 }}}{2} + \frac{\left. x^{3} - 2 x \right|_{\substack{ x=0 }}}{2} = 2 \]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 (-2, 0).✓ 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 claims to find 'every' number c, but only identifies one solution. The equation 3c^2 - 2 = 2 has two solutions, c = ±2√3/3. While c = 2√3/3 is outside the interval, the solution fails to explicitly check or reject the positive root, leaving the claim of finding 'every' number unsupported.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-01 — The solution claims to find 'every' number c, but only identifies one solution. The equation 3c^2 - 2 = 2 has two solutions, c = ±2√3/3. While c = 2√3/3 is outside the interval, the solution fails to explicitly check or reject the positive root, leaving the claim of finding 'every' number unsupported.gpt-oss:20b: pass 2026-11-01qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution fails to verify that the found value c lies within the open interval (-2, 0). Since c = -2*sqrt(3)/3 is approximately -1.15, it is in the interval, but the text claims it solves f'(c)=2 without showing the derivation or explicitly checking the interval constraint as part of the verification process. More critically, the problem asks to find *every* number c. The equation 3c^2 - 2 = 2 yields c = +/- 2*sqrt(3)/3. The positive root is outside the interval, but the solution does not explicitly reject it or mention the existence of the other root, potentially misleading the student into thinking there is only one solution to the derivative equation globally or skipping the exclusion step.gpt-oss:20b: pass 2026-10-31qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution fails to verify that the found value c lies within the open interval (-2, 0). The value c = -2*sqrt(3)/3 is approximately -1.15, which is in the interval, but the text claims it solves f'(c)=2. However, f'(-2*sqrt(3)/3) = 3*(4/3) - 2 = 2, so the algebra is correct. Wait, let me re-read the prompt's constraints. The prompt says 'equation... checked by CAS'. I should not re-check algebra. But I must check the sentences. Sentence 4 says 'c = ... solves ... and lies in (-2, 0)'. This is a sentence. Is it true? Yes. Is the setup correct? Yes. Why would it fail? Ah, look at the problem statement: 'find every number c'. The derivative is 3x^2 - 2 = 2 => 3x^2 = 4 => x = +/- 2/sqrt(3). The positive root 2/sqrt(3) is approx 1.15, which is NOT in (-2, 0). The negative root is in (-2, 0). So there is only one solution. The solution finds the correct one. Is there a subtle error? Let's check the secant slope calculation in sentence 2. f(-2) = -8 + 4 = -4. f(0) = 0. Slope = (0 - (-4)) / (0 - (-2)) = 4/2 = 2. The equation says 'Subs.../2 + Subs.../2 = 2'. This is a weird way to write the slope formula (f(b)-f(a))/(b-a). It writes (f(a)+f(b))/2? No, it writes Subs(..., -2)/2 + Subs(..., 0)/2. f(-2)/2 + f(0)/2 = -4/2 + 0 = -2. This is NOT 2. The equation claims the result is 2, but the expression evaluates to -2. The CAS check might have been on a different expression or the user typed the equation wrong. But the prompt says 'equation... checked by CAS... do not re-check the algebra'. However, if the equation is factually wrong in its representation of the slope, is that an error in the sentence or the equation? The sentence is just the comment '-- The slope of the secant line.' The equation is marked 'unchecked' in the prompt description? No, it says 'equation, unchecked' in the text provided by the user, but the instructions say 'equation... checked by CAS... do not re-check'. This is a contradiction in the prompt format provided by the user vs the instructions. The instructions say 'Each line... is either an EQUATION... checked... or a SENTENCE'. The user provided '2. [equation, unchecked] ...'. This implies the CAS did NOT check it, or the user is mislabeling. If I assume the CAS checked it, then the CAS must have verified that the expression equals 2. But -2 != 2. So either the CAS is broken or the expression is wrong. If the expression is wrong, the 'sentence' part '-- The slope of the secant line' is misleading because the equation does not represent the slope correctly (it represents the average of the function values, not the slope). The slope is (f(b)-f(a))/(b-a). The expression is (f(a)+f(b))/2. These are not the same. Thus, the setup for finding the slope is incorrect. The value 2 is correct for the slope, but the equation shown does not calculate the slope. This is a 'misleading' or 'error' in the modeling/setup. A student would learn that the slope is the average of the y-values, which is false.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.