Making a piecewise function continuous
Problem 1.404 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x + 5 & x < 2 \\ 3 x^{2} - 3 x - 2 & x \ge 2 \end{cases} \) 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.
- \[ \lim_{x \to 2^-}\left(k x + 5\right) = 2 k + 5 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 2^+}\left(3 x^{2} - 3 x - 2\right) = 4 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 4 \]Setting them equal gives k = -1/2.✓ Proved
Answer \( k = - \frac{1}{2} \)
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 "4 = 4" instead of setting the left‑hand limit 2k+5 equal to the right‑hand limit 4 and solving for k. This could mislead a student into thinking the equality is trivial rather than requiring solving for k.qwen3.6:27b-mlx: fail (error) — The solution claims that setting the limits equal gives k = -1/2, but the equation shown is 4 = 4, which is a tautology and does not involve k. The correct step should equate 2k + 5 to 4.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-07 — Sentence 4 incorrectly states "4 = 4" instead of setting the left‑hand limit 2k+5 equal to the right‑hand limit 4 and solving for k. This could mislead a student into thinking the equality is trivial rather than requiring solving for k.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution claims that setting the limits equal gives k = -1/2, but the equation shown is 4 = 4, which is a tautology and does not involve k. The correct step should equate 2k + 5 to 4.gpt-oss:20b: fail (error) 2026-10-07 — Sentence 4 incorrectly sets 4 = 4 instead of equating the left‑hand limit 2k+5 with the right‑hand limit 4. The conclusion that k = –1/2 is correct, but the justification is missing the necessary equation 2k+5 = 4.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution claims that setting the limits equal gives k = -1/2, but the equation shown is '4 = 4', which is a tautology and does not involve k. The correct step should equate 2k + 5 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 2026-10-07 with SymPy 1.14.0.