Home›Calculus 1›Making a piecewise function continuous›Problem 1.496

Making a piecewise function continuous

Problem 1.496 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k - 3 x^{2} & x < -2 \\ - 2 x - 4 & x \ge -2 \end{cases} \) 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
\[ \lim_{x \to -2^-}\left(k - 3 x^{2}\right) = k - 12 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to -2^+}\left(- 2 x - 4\right) = 0 \]
The right-hand limit, which is also f(c).✓ Proved
\[ 0 \]
Setting them equal gives k = 12.✓ Proved

Answer \( k = 12 \)

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: Each piece is a polynomial, so f is continuous away from x = -2. Continuity at c requires the left limit, right limit and f(c) to agree, which is the standard condition.
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 "0 = 0" instead of equating the left‑hand limit k‑12 to the right‑hand limit 0, which would yield k = 12.
qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal gives k = 12, but the equation shown is 0 = 0, which is a tautology and does not involve k. The correct equation should be k - 12 = 0.

Senior review claude-sonnet-5-5, 2026-10-09: fail — The answer k = 12 is correct, since k - 12 = 0 from matching the limits. The final line shows 0 = 0 rather than k - 12 = 0, so the key step is missing and a student would see a tautology presented as the derivation of k. The fix is to write k - 12 = 0 and then k = 12.

qwen3.6:27b-mlx: uphold — Line 4 displays only the tautology 0 = 0, which has no k in it; the equation k - 12 = 0 that actually produces k = 12 is never written, so the narration does not match the displayed line.

Every verdict on record (5)

gpt-oss:20b: fail (error) 2026-10-09 — Sentence 4 incorrectly states "0 = 0" instead of equating the left‑hand limit k‑12 to the right‑hand limit 0, which would yield k = 12.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 claims that setting the limits equal gives k = 12, but the equation shown is 0 = 0, which is a tautology and does not involve k. The correct equation should be k - 12 = 0.
claude-sonnet-5-5: fail (misleading) 2026-10-09 — The answer k = 12 is correct, since k - 12 = 0 from matching the limits. The final line shows 0 = 0 rather than k - 12 = 0, so the key step is missing and a student would see a tautology presented as the derivation of k. The fix is to write k - 12 = 0 and then k = 12.
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution claims that setting the limits equal gives k = 12, but the equation shown is '0 = 0', which is a tautology and does not involve k. The correct equation should be k - 12 = 0.

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 2026-10-09 with SymPy 1.14.0.