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Average value of a function

Problem 4.292 · medium

Find the average value of \( \displaystyle f(x) = \frac{4}{x^{2}} \) on \( \displaystyle [1, 3] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

\[ \int\limits_{1}^{3} \frac{4}{x^{2}}\, dx = \frac{8}{3} \]
The integral over the interval.✓ Proved
\[ \frac{4}{3} \]
Divide by the length of the interval.✓ Proved
\[ \frac{4}{3} \]
c = sqrt(3) lies in [1, 3].✓ Proved

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

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 calculate the average value using the formula (1/(b-a)) * integral, instead presenting tautological equations. It also asserts the value of c without showing the algebraic step of solving f(c) = f_ave.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly calculate the average value using the formula (1/(b-a)) * integral, instead presenting tautological equations. It also asserts the value of c without showing the algebraic step of solving f(c) = f_ave.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly state the formula for the average value, making the transition from the integral to the average value opaque. Furthermore, step 3 is logically incoherent: it presents an identity '4/3 = 4/3' while claiming to verify that c = sqrt(3) lies in the interval, which are unrelated statements.
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.