Home›Calculus 1›The Mean Value Theorem and Rolle's theorem›Problem 3.596

The Mean Value Theorem and Rolle's theorem

Problem 3.596 · medium

Verify that \( \displaystyle f(x) = - 2 x^{2} - 2 \) 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.
\[ - \frac{\left. - 2 x^{2} - 2 \right|_{\substack{ x=-2 }}}{2} + \frac{\left. - 2 x^{2} - 2 \right|_{\substack{ x=0 }}}{2} = 4 \]
The slope of the secant line.✓ Proved
\[ \frac{d}{d x} \left(- 2 x^{2} - 2\right) = - 4 x \]
Differentiate.✓ Proved
\[ 4 \]
c = -1 solves f′(c) = 4 and lies in (-2, 0).✓ Proved

Answer \( c = -1 \)

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 explicitly verify that the found value c = -1 lies within the open interval (-2, 0), which is a required part of the Mean Value Theorem conclusion. Additionally, the final line '4 = 4' is not a valid mathematical derivation showing that c = -1 is the solution; it merely states the result of substituting c=-1 into the derivative equation without showing the solving step.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly verify that the found value c = -1 lies within the open interval (-2, 0), which is a required part of the Mean Value Theorem conclusion. Additionally, the final line '4 = 4' is not a valid mathematical derivation showing that c = -1 is the solution; it merely states the result of substituting c=-1 into the derivative equation without showing the solving step.
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution claims that c = -1 solves f'(c) = 4, but f'(-1) = 4, so this part is actually correct. However, the setup in line 2 calculates the average rate of change as 4, which is correct: (f(0)-f(-2))/(0-(-2)) = (-2 - (-10))/2 = 8/2 = 4. Line 4 states '4 = 4' and claims c=-1 solves it. While the arithmetic is correct, the sentence in line 4 is poorly phrased and skips the step of solving -4c = 4 to get c = -1. More importantly, the prompt asks to verify hypotheses AND find c. The solution does find c=-1. Is there an error? Let's re-read carefully. f(x) = -2x^2 - 2. f(-2) = -2(4)-2 = -10. f(0) = -2. Slope = (-2 - (-10)) / (0 - (-2)) = 8 / 2 = 4. f'(x) = -4x. Set -4c = 4 => c = -1. -1 is in (-2, 0). The solution is mathematically correct. Why would it fail? Let's look at the 'unchecked' equations. Line 2: `Subs(-2*x**2 - 2, x, -2)/2 + Subs(-2*x**2 - 2, x, 0)/2 = 4`. This formula is WRONG. The slope of the secant line is `(f(b) - f(a)) / (b - a)`. The expression shown is `(f(a)/2 + f(b)/2)`. This is the average of the function values, not the slope. For this specific problem, f(-2)=-10, f(0)=-2. Average = (-10 + -2)/2 = -6. The equation claims it equals 4. So the equation in line 2 is FALSE. The computer algebra system marked it 'unchecked' in the prompt description, but the text says 'computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. Wait, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. It says 'do not re-check the algebra of a line marked proved'. Line 2 is marked 'unchecked' in the text provided by the user? No, the user text says `[equation, unchecked]`. This implies the CAS did NOT check it, or the user is presenting it as unchecked. But the instructions say 'do not re-check the algebra of a line marked proved'. Line 2 is NOT marked proved. It is marked unchecked. Therefore, I SHOULD check it. The formula for the slope is wrong. The value 4 is correct for the slope, but the expression `Subs(...)/2 + Subs(...)/2` evaluates to -6, not 4. So the equation is false. This is an error.
gpt-oss:20b: pass 2026-10-12

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 2026-10-12 with SymPy 1.14.0.