Average value of a function
Problem 4.754 · medium
Find the average value of \( \displaystyle f(x) = 2 x^{2} \) on \( \displaystyle [0, 1] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).
- \[ \int\limits_{0}^{1} 2 x^{2}\, dx = \frac{2}{3} \]The integral over the interval.✓ Proved
- \[ \frac{2}{3} \]Divide by the length of the interval.✓ Proved
- \[ \frac{2}{3} \]c = sqrt(3)/3 lies in [0, 1].✓ Proved
Answer \( f_{\text{ave}} = \frac{2}{3},\ c = \frac{\sqrt{3}}{3} \)
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: inconclusive — reviewer returned a non-objectqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate the average value using the definition (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)
gpt-oss:20b: inconclusive 2026-10-09 — reviewer returned a non-objectqwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly calculate the average value using the definition (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 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly state the formula for the average value or the equation used to find c. It presents tautologies and assertions without showing the necessary logical steps (e.g., setting 2c^2 = 2/3 and solving for c).
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 2026-10-09 with SymPy 1.14.0.