Average value of a function
Problem 4.601 · medium
Find the average value of \( \displaystyle f(x) = 3 x^{2} \) on \( \displaystyle [0, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).
- \[ \int\limits_{0}^{2} 3 x^{2}\, dx = 8 \]The integral over the interval.✓ Proved
- \[ 4 \]Divide by the length of the interval.✓ Proved
- \[ 4 \]c = 2*sqrt(3)/3 lies in [0, 2].✓ Proved
Answer \( f_{\text{ave}} = 4,\ c = \frac{2 \sqrt{3}}{3} \)
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: fail (error) — The solution never verifies that f(c)=f_ave; it only states that c lies in the interval. The conclusion that c is a point where f(c)=f_ave is unsupported.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the formula for the average value or the Mean Value Theorem for Integrals, making the logical steps opaque. Furthermore, line 2 is a tautology that does not demonstrate the calculation of the average value, and line 3 asserts the result without showing the derivation from f(c) = f_ave.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — The solution never verifies that f(c)=f_ave; it only states that c lies in the interval. The conclusion that c is a point where f(c)=f_ave is unsupported.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly state the formula for the average value or the Mean Value Theorem for Integrals, making the logical steps opaque. Furthermore, line 2 is a tautology that does not demonstrate the calculation of the average value, and line 3 asserts the result without showing the derivation from f(c) = f_ave.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to show the calculation of the average value or the derivation of c. It presents tautologies ('4=4') instead of the necessary steps: dividing the integral by the interval length to get f_ave, and solving 3c^2 = 4 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 2026-10-07 with SymPy 1.14.0.