Home›Calculus 1›The Mean Value Theorem and Rolle's theorem›Problem 3.400

The Mean Value Theorem and Rolle's theorem

Problem 3.400 · medium

Verify that \( \displaystyle f(x) = x^{3} - 5 x \) satisfies the hypotheses of the Mean Value Theorem on \( \displaystyle [0, 3] \), 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.
\[ - \frac{\left. x^{3} - 5 x \right|_{\substack{ x=0 }}}{3} + \frac{\left. x^{3} - 5 x \right|_{\substack{ x=3 }}}{3} = 4 \]
The slope of the secant line.✓ Proved
\[ \frac{d}{d x} \left(x^{3} - 5 x\right) = 3 x^{2} - 5 \]
Differentiate.✓ Proved
\[ 4 \]
c = sqrt(3) solves f′(c) = 4 and lies in (0, 3).✓ Proved

Answer \( c = \sqrt{3} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to find all values of c guaranteed by the Mean Value Theorem. The equation 3c^2 - 5 = 4 yields c = ±√3, but only c = √3 is in the interval (0, 3). While the final answer is correct, the step claiming 'c = sqrt(3) solves...' implies uniqueness or skips the negative root check without justification, which is misleading for a 'find every number' problem. More critically, the sentence in step 4 is not a logical derivation but a verification of a single candidate, failing to demonstrate how the candidate was found or that it is the only one.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to find all values of c guaranteed by the Mean Value Theorem. The equation 3c^2 - 5 = 4 yields c = ±√3, but only c = √3 is in the interval (0, 3). While the final answer is correct, the step claiming 'c = sqrt(3) solves...' implies uniqueness or skips the negative root check without justification, which is misleading for a 'find every number' problem. More critically, the sentence in step 4 is not a logical derivation but a verification of a single candidate, failing to demonstrate how the candidate was found or that it is the only one.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to solve for c explicitly. It states that c = sqrt(3) solves the equation but does not show the derivation from 3c^2 - 5 = 4, nor does it check for other solutions (like -sqrt(3)) and reject them based on the interval. This skips the core algebraic step required by the problem.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.