Making a piecewise function continuous
Problem 1.493 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x - 2 & x < 1 \\ 3 x^{2} - x + 2 & x \ge 1 \end{cases} \) 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.
- \[ \lim_{x \to 1^-}\left(k x - 2\right) = k - 2 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 1^+}\left(3 x^{2} - x + 2\right) = 4 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 4 \]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 (error) — Sentence 4 incorrectly sets the limits equal; it should equate k-2 to 4, yielding k=6, not 4=4.qwen3.6:27b-mlx: fail (error) — The solution sets up the equation 4 = 4 instead of equating the left-hand limit (k - 2) to the right-hand limit (4). This fails to solve for k and contains a logical contradiction in the derivation.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — Sentence 4 incorrectly sets the limits equal; it should equate k-2 to 4, yielding k=6, not 4=4.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution sets up the equation 4 = 4 instead of equating the left-hand limit (k - 2) to the right-hand limit (4). This fails to solve for k and contains a logical contradiction in the derivation.gpt-oss:20b: fail (error) 2026-10-09 — Sentence 4 incorrectly states 4 = 4 instead of equating the left‑hand limit k‑2 to the right‑hand limit 4. The correct conclusion is k‑2 = 4, yielding k = 6.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution claims that setting the limits equal gives k=6, but the equation shown is '4 = 4', which is a tautology and does not involve k. The correct step should equate k - 2 to 4.
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.