Average value of a function
Problem 4.757 · medium
Find the average value of \( \displaystyle f(x) = x^{3} - x \) on \( \displaystyle [0, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).
- \[ \int\limits_{0}^{2} \left(x^{3} - x\right)\, dx = 2 \]The integral over the interval.✓ Proved
- \[ 1 \]Divide by the length of the interval.✓ Proved
- \[ - \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + \left(\frac{1}{3 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}\right)^{3} = 1 \]c = 1/(3*(sqrt(69)/18 + 1/2)**(1/3)) + (sqrt(69)/18 + 1/2)**(1/3) lies in [0, 2].✓ Proved
Answer \( f_{\text{ave}} = 1,\ c = \frac{1}{3 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}} \)
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: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-object
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.