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Average value of a function

Problem 4.597 · medium

Find the average value of \( \displaystyle f(x) = 3 \sqrt{x} \) on \( \displaystyle [0, 9] \), and every \( \displaystyle c \) in the interval with \( \displaystyle f(c) = f_{\text{ave}} \).

\[ \int\limits_{0}^{9} 3 \sqrt{x}\, dx = 54 \]
The integral over the interval.✓ Proved
\[ 6 \]
Divide by the length of the interval.✓ Proved
\[ 6 \]
c = 4 lies in [0, 9].✓ Proved

Answer \( f_{\text{ave}} = 6,\ c = 4 \)

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: inconclusive — reviewer returned a non-object
qwen3.6:27b-mlx: fail (error) — The solution fails to state the definition of average value or the Mean Value Theorem for Integrals, making the jump from the integral result to the final answer unjustified. It also omits the crucial step of solving f(c) = f_ave to find c.

Every verdict on record (4)

gpt-oss:20b: inconclusive 2026-10-07 — reviewer returned a non-object
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to state the definition of average value or the Mean Value Theorem for Integrals, making the jump from the integral result to the final answer unjustified. It also omits the crucial step of solving f(c) = f_ave to find c.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution lacks the explicit setup for the average value formula and the Mean Value Theorem for Integrals. It presents arithmetic checks without showing the derivation of f_ave or the solution for c, making the logic opaque and incomplete.

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 2026-10-07 with SymPy 1.14.0.