Making a piecewise function continuous
Problem 1.494 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} + 1 & x < 1 \\ - 3 x^{2} - x - 1 & x \ge 1 \end{cases} \) 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.
- \[ \lim_{x \to 1^-}\left(k x^{2} + 1\right) = k + 1 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 1^+}\left(- 3 x^{2} - x - 1\right) = -5 \]The right-hand limit, which is also f(c).✓ Proved
- \[ -5 \]Setting them equal gives k = -6.✓ Proved
Answer \( k = -6 \)
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 (misleading) — Sentence 4 incorrectly states "-5 = -5" instead of equating the left- and right-hand limits (k+1 = -5). This misleads a student into thinking the equality is trivial, hiding the actual step that yields k = -6.qwen3.6:27b-mlx: fail (error) — The solution sets the right-hand limit (-5) equal to itself (-5) instead of equating it to the left-hand limit (k + 1). This fails to solve for k and results in a tautology rather than the required equation k + 1 = -5.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-09 — Sentence 4 incorrectly states "-5 = -5" instead of equating the left- and right-hand limits (k+1 = -5). This misleads a student into thinking the equality is trivial, hiding the actual step that yields k = -6.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution sets the right-hand limit (-5) equal to itself (-5) instead of equating it to the left-hand limit (k + 1). This fails to solve for k and results in a tautology rather than the required equation k + 1 = -5.gpt-oss:20b: fail (misleading) 2026-10-09 — The fourth sentence incorrectly states the equality as "-5 = -5" instead of "k + 1 = -5", which could confuse a student about how the value of k is determined.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution claims that setting the limits equal yields k = -6, but the equation shown is '-5 = -5', which is a tautology and does not involve k. The correct equation should be 'k + 1 = -5'.
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.