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Making a piecewise function continuous

Problem 1.292 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k - x^{2} & x < 3 \\ 3 - 4 x & x \ge 3 \end{cases} \) continuous for all \( \displaystyle x \).

Each piece is a polynomial, so f is continuous everywhere except possibly at x = 3. There the left-hand limit, the right-hand limit and f(c) must agree.
Reviewed
\[ \lim_{x \to 3^-}\left(k - x^{2}\right) = k - 9 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to 3^+}\left(3 - 4 x\right) = -9 \]
The right-hand limit, which is also f(c).✓ Proved
\[ -9 \]
Setting them equal gives k = 0.✓ Proved

Answer \( k = 0 \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: A sentence stating the continuity setup: polynomial pieces are continuous away from x=3, and at x=3 the one-sided limits must equal f(3). This is the standard definition and is correct.
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	with that k the two pieces agree at c ± 1e-10

Reviewers

gpt-oss:20b: fail (error) — Sentence 4 incorrectly equates the two limits as -9 = -9, omitting the left‑hand limit expression k-9. The correct equation is k-9 = -9, which yields k = 0.
qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal gives k = 0, but the equation derived is k - 9 = -9, which implies k = 9. The stated answer k = 0 is incorrect.

Senior review claude-sonnet-5-5, 2026-10-05: pass — The math is correct: the left limit is k-9, the right limit and f(3) are -9, and k=0. Line 4 shows only the value -9 = -9 (the equation with k=0 substituted) rather than the equation k-9 = -9, so the solving step is compressed, but nothing is false.

qwen3.6:27b-mlx: dismiss — Solving k - 9 = -9 gives k = -9 + 9 = 0, not 9, so the stated answer k = 0 is correct and the objection's arithmetic is wrong.

Every verdict on record (5)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 4 claims that setting the limits equal gives k = 0, but the equation derived is k - 9 = -9, which implies k = 9. The stated answer k = 0 is incorrect.
gpt-oss:20b: fail (error) 2026-10-05 — Sentence 4 incorrectly equates the two limits as -9 = -9, omitting the left‑hand limit expression k-9. The correct equation is k-9 = -9, which yields k = 0.
claude-sonnet-5-5: pass 2026-10-05 — The math is correct: the left limit is k-9, the right limit and f(3) are -9, and k=0. Line 4 shows only the value -9 = -9 (the equation with k=0 substituted) rather than the equation k-9 = -9, so the solving step is compressed, but nothing is false.
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution sets the left-hand limit (k - 9) equal to the right-hand limit (-9) but incorrectly concludes k = 0. Solving k - 9 = -9 yields k = 9.
gpt-oss:20b: pass 2026-10-05

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/continuity_parameter, checked 2026-10-05 with SymPy 1.14.0.