Making a piecewise function continuous
Problem 1.299 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k - x^{2} & x < -1 \\ 3 - 4 x & x \ge -1 \end{cases} \) continuous for all \( \displaystyle x \).
- Each piece is a polynomial, so f is continuous everywhere except possibly at x = -1. There the left-hand limit, the right-hand limit and f(c) must agree.
- \[ \lim_{x \to -1^-}\left(k - x^{2}\right) = k - 1 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to -1^+}\left(3 - 4 x\right) = 7 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 7 \]Setting them equal gives k = 8.✓ Proved
Answer \( k = 8 \)
Lines: 3 proved, 1 not checked. 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 | Not checked | — | a sentence; read, not computed |
| 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 sets 7=7 instead of equating the two limits k-1 and 7 to solve for k. The correct conclusion is k-1=7, yielding k=8.qwen3.6:27b-mlx: fail (error) — Line 4 claims that setting the limits equal gives k=8, but the equation shown is '7 = 7', which is a tautology and does not involve k. The correct step should equate the left-hand limit (k-1) to the right-hand limit (7), i.e., k - 1 = 7.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Line 4 claims that setting the limits equal gives k=8, but the equation shown is '7 = 7', which is a tautology and does not involve k. The correct step should equate the left-hand limit (k-1) to the right-hand limit (7), i.e., k - 1 = 7.gpt-oss:20b: fail (error) 2026-10-05 — Sentence 4 incorrectly sets 7=7 instead of equating the two limits k-1 and 7 to solve for k. The correct conclusion is k-1=7, yielding k=8.qwen3.6:27b-mlx: fail (error) 2026-10-05 — Line 4 claims that setting the limits equal (k - 1 = 7) results in the equation 7 = 7, which is algebraically incorrect and fails to show the step k = 8.gpt-oss:20b: fail (misleading) 2026-10-05 — The solution states "7 = 7" after equating the limits, which is incorrect. The correct equation is k - 1 = 7, leading to k = 8. The conclusion is misleading because it does not show the proper algebraic step to find k.
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.