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Making a piecewise function continuous

Problem 1.405 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} - 1 & x < -2 \\ - 3 x^{2} + 2 x - 1 & x \ge -2 \end{cases} \) continuous for all \( \displaystyle x \).

Each piece is a polynomial, so f is continuous everywhere except possibly at x = -2. There the left-hand limit, the right-hand limit and f(c) must agree.
Reviewed
\[ \lim_{x \to -2^-}\left(k x^{2} - 1\right) = 4 k - 1 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to -2^+}\left(- 3 x^{2} + 2 x - 1\right) = -17 \]
The right-hand limit, which is also f(c).✓ Proved
\[ -17 \]
Setting them equal gives k = -4.✓ Proved

Answer \( k = -4 \)

Lines: 3 proved, 1 reviewed. 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	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: This is a sentence, not an algebraic equivalence. It correctly says each piece is a polynomial, hence continuous, so the only possible discontinuity is at x = -2, where the left limit, right limit and f(-2) must agree.
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 states that the limits are equal as "-17 = -17" instead of solving 4k-1 = -17 to find k = -4. The conclusion is not justified by the previous lines.
qwen3.6:27b-mlx: fail (error) — Line 4 is mathematically incorrect; it claims that setting the limits equal (-17 = 4k - 1) results in the identity -17 = -17, rather than solving for k. The correct step would be 4k - 1 = -17, which yields k = -4.

Senior review claude-sonnet-5-5, 2026-10-07: fail — The answer k = -4 is correct (4k - 1 = -17 gives k = -4, and the right-hand limit is -17). However, the final line is a vacuous identity that does not show k being determined. The solution should state 4k - 1 = -17 and then solve it for k.

qwen3.6:27b-mlx: uphold — Line 4 is the tautology -17 = -17 with no k in it, yet it is described as 'setting them equal gives k = -4'; the needed equation 4k - 1 = -17 and its solution never appear, so the derivation of k is missing.

Every verdict on record (5)

gpt-oss:20b: fail (error) 2026-10-07 — Sentence 4 incorrectly states that the limits are equal as "-17 = -17" instead of solving 4k-1 = -17 to find k = -4. The conclusion is not justified by the previous lines.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — Line 4 is mathematically incorrect; it claims that setting the limits equal (-17 = 4k - 1) results in the identity -17 = -17, rather than solving for k. The correct step would be 4k - 1 = -17, which yields k = -4.
claude-sonnet-5-5: fail (misleading) 2026-10-07 — The answer k = -4 is correct (4k - 1 = -17 gives k = -4, and the right-hand limit is -17). However, the final line is a vacuous identity that does not show k being determined. The solution should state 4k - 1 = -17 and then solve it for k.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 4 claims that setting the limits equal gives k = -4, but the equation shown is -17 = -17, which is a tautology and does not involve k. The correct equation should be 4k - 1 = -17.

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 2026-10-07 with SymPy 1.14.0.