The Mean Value Theorem and Rolle's theorem
Problem 3.547 · medium
Verify that \( \displaystyle f(x) = 3 x^{2} + 3 \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [1, 3] \), 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.
- \[ - \frac{\left. 3 x^{2} + 3 \right|_{\substack{ x=1 }}}{2} + \frac{\left. 3 x^{2} + 3 \right|_{\substack{ x=3 }}}{2} = 12 \]The slope of the secant line.✓ Proved
- \[ \frac{d}{d x} \left(3 x^{2} + 3\right) = 6 x \]Differentiate.✓ Proved
- \[ 12 \]c = 2 solves f′(c) = 12 and lies in (1, 3).✓ Proved
Answer \( c = 2 \)
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) — Step 2 calculates the average of the function values f(1) and f(3) instead of the slope of the secant line, which is (f(3) - f(1)) / (3 - 1). Although the numerical result 12 is coincidentally correct, the formula stated is wrong and would fail for non-symmetric intervals or functions.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-10 — Step 2 calculates the average of the function values f(1) and f(3) instead of the slope of the secant line, which is (f(3) - f(1)) / (3 - 1). Although the numerical result 12 is coincidentally correct, the formula stated is wrong and would fail for non-symmetric intervals or functions.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 explicitly solve the equation f'(c) = 12 for c, jumping directly to the verification that c=2 works. While the final answer is correct, the step '12 = 12' does not demonstrate finding the value, only checking it, which is logically incomplete for a 'find' problem.
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.