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Average value of a function

Problem 4.898 · medium

Find the average value of \( \displaystyle f(x) = 4 x^{2} \) on \( \displaystyle [0, 1] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

\[ \int\limits_{0}^{1} 4 x^{2}\, dx = \frac{4}{3} \]
The integral over the interval.✓ Proved
\[ \frac{4}{3} \]
Divide by the length of the interval.✓ Proved
\[ \frac{4}{3} \]
c = sqrt(3)/3 lies in [0, 1].✓ Proved

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

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. A reviewer's objection to the explanation stands; see 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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution never applies the average value formula \n\[\overline{f}=\frac{1}{b-a}\int_a^b f(x)\,dx\] for the interval \([0,1]\). It simply states \"4/3
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the formula for the average value or the Mean Value Theorem for Integrals, making the logical steps opaque. Furthermore, line 3 asserts a result without showing the derivation (solving 4c^2 = 4/3), which is the core of the second part of the problem.

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 the Mean Value Theorem for Integrals, making the logical steps opaque. Furthermore, line 3 asserts a result without showing the derivation (solving 4c^2 = 4/3), which is the core of the second part of the problem.
gpt-oss:20b: inconclusive 2026-10-11 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution never applies the average value formula \n\[\overline{f}=\frac{1}{b-a}\int_a^b f(x)\,dx\] for the interval \([0,1]\). It simply states \"4/3
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly state the formula for the average value or the equation f(c) = f_ave. Line 2 is a tautology that does not show the division by interval length, and Line 3 asserts the result without showing the algebraic step of solving 4c^2 = 4/3.
gpt-oss:20b: fail (misleading) 2026-10-11 — The solution omits the crucial step that f(c)=f_ave; it only states that c lies in the interval, which is insufficient for a correct application of the Mean Value Theorem for Integrals.

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.