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Average value of a function

Problem 4.293 · medium

Find the average value of \( \displaystyle f(x) = 2 x^{2} - 2 x + 3 \) on \( \displaystyle [-1, 2] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

\[ \int\limits_{-1}^{2} \left(2 x^{2} - 2 x + 3\right)\, dx = 12 \]
The integral over the interval.✓ Proved
\[ 4 \]
Divide by the length of the interval.✓ Proved
\[ 2 \left(\frac{1}{2} - \frac{\sqrt{3}}{2}\right)^{2} + \sqrt{3} + 2 = 4 \]
c = 1/2 - sqrt(3)/2 lies in [-1, 2].✓ Proved
\[ - \sqrt{3} + 2 + 2 \left(\frac{1}{2} + \frac{\sqrt{3}}{2}\right)^{2} = 4 \]
c = 1/2 + sqrt(3)/2 lies in [-1, 2].✓ Proved

Answer \( f_{\text{ave}} = 4,\ c = \frac{1}{2} - \frac{\sqrt{3}}{2},\ \frac{1}{2} + \frac{\sqrt{3}}{2} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to show the derivation of the values for c. It merely verifies that the stated answers satisfy the equation f(c) = 4, but does not demonstrate the process of solving 2x^2 - 2x + 3 = 4 to find these roots, which is the core of the second part of the problem.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to show the derivation of the values for c. It merely verifies that the stated answers satisfy the equation f(c) = 4, but does not demonstrate the process of solving 2x^2 - 2x + 3 = 4 to find these roots, which is the core of the second part of the problem.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly calculate the average value using the definition (1/(b-a) * integral), instead jumping to the result. It also fails to show the derivation of the values for c by solving f(c) = f_ave, merely verifying the final answers.
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.