Average value of a function
Problem 4.900 · medium
Find the average value of \( \displaystyle f(x) = 1 - 3 x^{2} \) on \( \displaystyle [-2, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).
- \[ \int\limits_{-2}^{2} \left(1 - 3 x^{2}\right)\, dx = -12 \]The integral over the interval.✓ Proved
- \[ -3 \]Divide by the length of the interval.✓ Proved
- \[ -3 \]c = -2*sqrt(3)/3 lies in [-2, 2].✓ Proved
- \[ -3 \]c = 2*sqrt(3)/3 lies in [-2, 2].✓ Proved
Answer \( f_{\text{ave}} = -3,\ c = - \frac{2 \sqrt{3}}{3},\ \frac{2 \sqrt{3}}{3} \)
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 |
| 4 | ✓ 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 formula for the average value or show the division by the interval length (4), making the jump from the integral (-12) to the average (-3) unjustified in the text. Furthermore, it does not show the algebraic steps solving f(c) = -3 to find the values of c.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly state the formula for the average value or show the division by the interval length (4), making the jump from the integral (-12) to the average (-3) unjustified in the text. Furthermore, it does not show the algebraic steps solving f(c) = -3 to find the values of c.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to show the calculation of the average value (integral divided by interval length) and the derivation of c from f(c) = f_ave. It merely asserts the final numbers without demonstrating the required steps.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.