Average value of a function
Problem 4.297 · medium
Find the average value of \( \displaystyle f(x) = - 3 x^{2} - x \) on \( \displaystyle [-2, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).
- \[ \int\limits_{-2}^{2} \left(- 3 x^{2} - x\right)\, dx = -16 \]The integral over the interval.✓ Proved
- \[ -4 \]Divide by the length of the interval.✓ Proved
- \[ -4 \]c = -4/3 lies in [-2, 2].✓ Proved
- \[ -4 \]c = 1 lies in [-2, 2].✓ Proved
Answer \( f_{\text{ave}} = -4,\ c = - \frac{4}{3},\ 1 \)
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: fail (misleading) — The solution states that c = -4/3 and c = 1 lie in the interval, but it does not actually verify that f(c) equals the average value. The equations shown merely repeat -4 = -4 without computing f(c), which could mislead a student into thinking the condition f(c)=f_ave has been proved.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the formula for the average value, making the division in step 2 unjustified. Furthermore, steps 3 and 4 are logically incoherent; they equate the average value to itself rather than verifying that f(c) equals the average value, and they do not show the derivation of the c values.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly state the formula for the average value, making the division in step 2 unjustified. Furthermore, steps 3 and 4 are logically incoherent; they equate the average value to itself rather than verifying that f(c) equals the average value, and they do not show the derivation of the c values.gpt-oss:20b: fail (misleading) 2026-10-04 — The solution states that c = -4/3 and c = 1 lie in the interval, but it does not actually verify that f(c) equals the average value. The equations shown merely repeat -4 = -4 without computing f(c), which could mislead a student into thinking the condition f(c)=f_ave has been proved.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to show the calculation of the average value or the derivation of the values for c. It merely asserts the final answers without demonstrating the necessary steps (integrating, dividing by interval length, and solving the quadratic equation).gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.