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Average value of a function

Problem 4.903 · medium

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

\[ \int\limits_{-1}^{2} \left(- x^{2} + x - 2\right)\, dx = - \frac{15}{2} \]
The integral over the interval.✓ Proved
\[ - \frac{5}{2} \]
Divide by the length of the interval.✓ Proved
\[ - \frac{3}{2} - \frac{\sqrt{3}}{2} - \left(\frac{1}{2} - \frac{\sqrt{3}}{2}\right)^{2} = - \frac{5}{2} \]
c = 1/2 - sqrt(3)/2 lies in [-1, 2].✓ Proved
\[ - \left(\frac{1}{2} + \frac{\sqrt{3}}{2}\right)^{2} - \frac{3}{2} + \frac{\sqrt{3}}{2} = - \frac{5}{2} \]
c = 1/2 + sqrt(3)/2 lies in [-1, 2].✓ Proved

Answer \( f_{\text{ave}} = - \frac{5}{2},\ c = \frac{1}{2} - \frac{\sqrt{3}}{2},\ \frac{1}{2} + \frac{\sqrt{3}}{2} \)

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 explicitly state the formula for the average value or show the division of the integral by the interval length (3). It also skips the step of setting f(c) equal to the average value to solve for c, instead just verifying the roots.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly state the formula for the average value or show the division of the integral by the interval length (3). It also skips the step of setting f(c) equal to the average value to solve for c, instead just verifying the roots.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly state the formula for the average value or show the calculation of the interval length, making the transition from the integral to the average value unjustified. Furthermore, the equations in steps 3 and 4 are malformed; they substitute the x-coordinate into the function but include extraneous terms (like '-3/2') that do not correspond to f(c), rendering the verification of the Mean Value Theorem points algebraically incorrect and confusing.
gpt-oss:20b: inconclusive 2026-10-11 — 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 2026-10-11 with SymPy 1.14.0.