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Average value of a function

Problem 4.896 · medium

Find the average value of \( \displaystyle f(x) = 2 x^{2} + 2 x + 1 \) on \( \displaystyle [-2, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

\[ \int\limits_{-2}^{2} \left(2 x^{2} + 2 x + 1\right)\, dx = \frac{44}{3} \]
The integral over the interval.✓ Proved
\[ \frac{11}{3} \]
Divide by the length of the interval.✓ Proved
\[ - \frac{\sqrt{57}}{3} + 2 \left(- \frac{\sqrt{57}}{6} - \frac{1}{2}\right)^{2} = \frac{11}{3} \]
c = -sqrt(57)/6 - 1/2 lies in [-2, 2].✓ Proved
\[ 2 \left(- \frac{1}{2} + \frac{\sqrt{57}}{6}\right)^{2} + \frac{\sqrt{57}}{3} = \frac{11}{3} \]
c = -1/2 + sqrt(57)/6 lies in [-2, 2].✓ Proved

Answer \( f_{\text{ave}} = \frac{11}{3},\ c = - \frac{\sqrt{57}}{6} - \frac{1}{2},\ - \frac{1}{2} + \frac{\sqrt{57}}{6} \)

Every line of this solution was proved by the computer algebra system SymPy. 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
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	numerical quadrature for the average, and f evaluated at each c

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the formula for the average value (integral divided by interval length) or set up the equation f(c) = f_ave to solve for c. It merely verifies the final numerical answers without showing the derivation, which is the core of the problem.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly state the formula for the average value (integral divided by interval length) or set up the equation f(c) = f_ave to solve for c. It merely verifies the final numerical answers without showing the derivation, which is the core of the problem.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly state the formula for the average value or show the division of the integral by the interval length (4). It jumps from the integral result to the final average value without the necessary logical step, making the derivation incomplete.
gpt-oss:20b: pass 2026-10-11

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/average_value, checked 2026-10-11 with SymPy 1.14.0.