Average value of a function
Problem 4.756 · medium
Find the average value of \( \displaystyle f(x) = 3 x^{2} \) on \( \displaystyle [0, 1] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).
- \[ \int\limits_{0}^{1} 3 x^{2}\, dx = 1 \]The integral over the interval.✓ Proved
- \[ 1 \]Divide by the length of the interval.✓ Proved
- \[ 1 \]c = sqrt(3)/3 lies in [0, 1].✓ Proved
Answer \( f_{\text{ave}} = 1,\ c = \frac{\sqrt{3}}{3} \)
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. A reviewer's objection to the explanation stands; see 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 state the definition of average value (integral divided by interval length) and the Mean Value Theorem for Integrals. It presents disjointed equations without logical connectives, making it unclear how the result is derived or why c = sqrt(3)/3 is the correct solution to 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 state the definition of average value (integral divided by interval length) and the Mean Value Theorem for Integrals. It presents disjointed equations without logical connectives, making it unclear how the result is derived or why c = sqrt(3)/3 is the correct solution to f(c) = f_ave.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 state the formula for the average value or the equation used to find c, making the logical steps opaque. Specifically, line 3 asserts that c = sqrt(3)/3 is in the interval without showing the derivation from f(c) = f_ave.
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.