Average value of a function
Problem 4.448 · medium
Find the average value of \( \displaystyle f(x) = \frac{2}{x^{2}} \) on \( \displaystyle [1, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).
- \[ \int\limits_{1}^{2} \frac{2}{x^{2}}\, dx = 1 \]The integral over the interval.✓ Proved
- \[ 1 \]Divide by the length of the interval.✓ Proved
- \[ 1 \]c = sqrt(2) lies in [1, 2].✓ Proved
Answer \( f_{\text{ave}} = 1,\ c = \sqrt{2} \)
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 |
| 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, $f_{\text{ave}} = \frac{1}{b-a} \int_a^b f(x) dx$, making the transition from the integral to the average value opaque. Furthermore, it does not show the step of solving $f(c) = f_{\text{ave}}$ to find $c$, merely asserting the result.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly state the formula for the average value, $f_{\text{ave}} = \frac{1}{b-a} \int_a^b f(x) dx$, making the transition from the integral to the average value opaque. Furthermore, it does not show the step of solving $f(c) = f_{\text{ave}}$ to find $c$, merely asserting the result.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly state the formula for the average value, making the division step (line 2) unjustified. It also skips the crucial step of setting f(c) equal to the average value to solve for c, jumping directly to the answer.gpt-oss:20b: fail (misleading) 2026-10-05 — The solution skips the crucial step of dividing the integral by the interval length to obtain the average value, and it does not verify that f(√2)=f_ave. The statements about the average value and the choice of c are therefore unsupported and could mislead a student.
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 2026-10-05 with SymPy 1.14.0.