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Average value of a function

Problem 4.296 · medium

Find the average value of \( \displaystyle f(x) = x^{2} \) on \( \displaystyle [0, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

\[ \int\limits_{0}^{2} x^{2}\, dx = \frac{8}{3} \]
The integral over the interval.✓ Proved
\[ \frac{4}{3} \]
Divide by the length of the interval.✓ Proved
\[ \frac{4}{3} \]
c = 2*sqrt(3)/3 lies in [0, 2].✓ Proved

Answer \( f_{\text{ave}} = \frac{4}{3},\ c = \frac{2 \sqrt{3}}{3} \)

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
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 definition of average value or the Mean Value Theorem for Integrals, making the logical connection between the integral calculation and the final answer opaque. Additionally, line 3 is a tautology that does not demonstrate that c is in the interval or that f(c) equals the average value.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly state the definition of average value or the Mean Value Theorem for Integrals, making the logical connection between the integral calculation and the final answer opaque. Additionally, line 3 is a tautology that does not demonstrate that c is in the interval or that f(c) equals the average value.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution lacks the explicit setup for the average value formula and the Mean Value Theorem for Integrals. It presents disjointed equations without stating the definitions or theorems used, failing to show that f(c) = f_ave was solved to find c.
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/average_value, checked 2026-10-04 with SymPy 1.14.0.