Home›Calculus 1›The Mean Value Theorem and Rolle's theorem›Problem 3.767

The Mean Value Theorem and Rolle's theorem

Problem 3.767 · medium

Verify that \( \displaystyle f(x) = - 4 x^{2} - 2 x + 3 \) 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.
Reviewed
\[ \frac{\left(-1\right) \left. - 4 x^{2} - 2 x + 3 \right|_{\substack{ x=0 }}}{3} + \frac{\left. - 4 x^{2} - 2 x + 3 \right|_{\substack{ x=3 }}}{3} = -14 \]
The slope of the secant line.✓ Proved
\[ \frac{d}{d x} \left(- 4 x^{2} - 2 x + 3\right) = - 8 x - 2 \]
Differentiate.✓ Proved
\[ \left. - 8 x - 2 \right|_{\substack{ x=\frac{3}{2} }} = -14 \]
c = 3/2 solves f′(c) = -14 and lies in (0, 3).✓ Proved

Answer \( c = \frac{3}{2} \)

Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies that polynomials satisfy the MVT hypotheses, calculates the secant slope accurately, and solves for c within the open interval.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly identifies that polynomials satisfy the MVT hypotheses, calculates the secant slope accurately, and solves for c within the open interval.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: fail (error) 2026-10-31 — The solution claims c = 3/2 is the solution, but f'(3/2) = -14 is false (f'(3/2) = -14 is incorrect; f'(3/2) = -8(1.5) - 2 = -12 - 2 = -14, wait, -14 is correct. Let me re-read. f'(c) = -14. -8c - 2 = -14 => -8c = -12 => c = 12/8 = 3/2. The calculation is actually correct. Why did I think it was wrong? Let me check the secant slope. f(3) = -4(9) - 2(3) + 3 = -36 - 6 + 3 = -39. f(0) = 3. Slope = (-39 - 3) / (3 - 0) = -42 / 3 = -14. The equation in line 2 is weirdly written but evaluates to -14. Line 4 checks out. Is there another c? f'(c) = -14 is linear, so only one solution. The solution seems correct. Wait, let me look closer at line 2. It says `Subs(..., 0)/3 + Subs(..., 3)/3`. This is (f(0) + f(3))/3. This is NOT the slope of the secant line. The slope is (f(3) - f(0)) / (3 - 0). The formula used is incorrect, even though the numerical result -14 happens to be correct by coincidence? Let's check: (3 + (-39))/3 = -36/3 = -12. Wait. Line 2 says the result is -14. But (f(0)+f(3))/3 = -12. So the equation in line 2 is FALSE. The CAS marked it 'proved', which implies the CAS might have evaluated it differently or the prompt implies the CAS checked the algebra of the expression shown. If the expression shown is `(f(0)+f(3))/3`, it equals -12, not -14. If the CAS says it equals -14, the CAS is wrong or the expression is not what is written. However, the instruction says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. This is a trap. I must trust the algebraic equality if marked proved? No, the prompt says 'do not re-check the algebra of a line marked proved'. This usually means if the line says `A = B` and it's marked proved, assume `A=B` is true. But here, the sentence part is 'The slope of the secant line'. The formula used to calculate the slope is wrong. The text claims the expression IS the slope. It is not. The expression `(f(0)+f(3))/3` is not the slope. The slope is `(f(3)-f(0))/3`. Even if the numbers coincidentally matched (they don't, -12 != -14), the method described is wrong. Actually, looking at the numbers: f(0)=3, f(3)=-39. Slope = -14. The expression in line 2 evaluates to -12. So the equation `... = -14` is factually false. But I am told not to re-check algebra. This is contradictory. If I cannot re-check algebra, I must assume the equation is true. But the sentence 'The slope of the secant line' describes the LHS. The LHS is NOT the slope. So the sentence is false. The setup for the MVT requires the slope of the secant line. The solution uses a wrong formula for the slope. This is an error in the method/setup description.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: fail (error) 2026-10-31 — The calculation of the secant slope in step 2 is incorrect; the formula used averages the function values instead of computing the difference quotient (f(b)-f(a))/(b-a). The correct slope is -14, but the derivation shown yields -14 only by coincidence of the specific numbers if interpreted loosely, yet the formula written is mathematically wrong for a slope.
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.