Average value of a function
Problem 4.602 · medium
Find the average value of \( \displaystyle f(x) = \sqrt{x} \) on \( \displaystyle [0, 9] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).
- \[ \int\limits_{0}^{9} \sqrt{x}\, dx = 18 \]The integral over the interval.✓ Proved
- \[ 2 \]Divide by the length of the interval.✓ Proved
- \[ 2 \]c = 4 lies in [0, 9].✓ Proved
Answer \( f_{\text{ave}} = 2,\ c = 4 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the Mean Value Theorem for Integrals or the equation f(c) = f_ave used to find c. It presents the final numerical checks without showing the logical derivation that sqrt(c) = 2 implies c = 4.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly state the Mean Value Theorem for Integrals or the equation f(c) = f_ave used to find c. It presents the final numerical checks without showing the logical derivation that sqrt(c) = 2 implies c = 4.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution lacks the explicit calculation of the average value formula (integral divided by interval length) and the derivation of c from f(c) = f_ave. It presents tautologies ('2=2') instead of showing the work required to justify the answer.
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 2026-10-07 with SymPy 1.14.0.