Average value of a function
Problem 4.594 · medium
Find the average value of \( \displaystyle f(x) = x^{3} - 2 x \) on \( \displaystyle [0, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).
- \[ \int\limits_{0}^{2} \left(x^{3} - 2 x\right)\, dx = 0 \]The integral over the interval.✓ Proved
- \[ 0 \]Divide by the length of the interval.✓ Proved
- \[ 0 \]c = 0 lies in [0, 2].✓ Proved
- \[ 0 \]c = sqrt(2) lies in [0, 2].✓ Proved
Answer \( f_{\text{ave}} = 0,\ c = 0,\ \sqrt{2} \)
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 |
| 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 calculate the average value using the formula f_ave = (1/(b-a)) * integral, instead just showing the integral is 0. It also fails to solve f(c) = f_ave to find all values of c, omitting c = -sqrt(2) is not in the interval but the logic for finding c is missing entirely, and it does not address if there are other solutions or justify why only 0 and sqrt(2) are listed (though -sqrt(2) is outside, the step of solving x^3-2x=0 is skipped).
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 calculate the average value using the formula f_ave = (1/(b-a)) * integral, instead just showing the integral is 0. It also fails to solve f(c) = f_ave to find all values of c, omitting c = -sqrt(2) is not in the interval but the logic for finding c is missing entirely, and it does not address if there are other solutions or justify why only 0 and sqrt(2) are listed (though -sqrt(2) is outside, the step of solving x^3-2x=0 is skipped).qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to calculate the average value, incorrectly stating the integral is 0 (it is 4) and dividing by 0 instead of the interval length 2. Consequently, the derived values for c are incorrect.gpt-oss:20b: pass 2026-10-07
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.