Making a piecewise function continuous
Problem 1.548 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k - 2 x^{2} & x < 3 \\ - 4 x - 4 & x \ge 3 \end{cases} \) continuous for all \( \displaystyle x \).
- Each piece is a polynomial, so f is continuous everywhere except possibly at x = 3. There the left-hand limit, the right-hand limit and f(c) must agree.
- \[ \lim_{x \to 3^-}\left(k - 2 x^{2}\right) = k - 18 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 3^+}\left(- 4 x - 4\right) = -16 \]The right-hand limit, which is also f(c).✓ Proved
- \[ -16 \]Setting them equal gives k = 2.✓ Proved
Answer \( k = 2 \)
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) — The solution incorrectly equates the right‑hand limit to itself instead of setting the left‑hand limit equal to the right‑hand limit. This would mislead a student into thinking no calculation is needed to find k, whereas the correct step is k-18 = -16, yielding k = 2.qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal gives k = 2, but the equation shown is -16 = -16, which is independent of k. The correct equation should be k - 18 = -16, which yields k = 2.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — Step 4 claims that setting the limits equal gives k = 2, but the equation shown is -16 = -16, which is independent of k. The correct equation should be k - 18 = -16, which yields k = 2.gpt-oss:20b: fail (misleading) 2026-10-11 — The solution incorrectly equates the right‑hand limit to itself instead of setting the left‑hand limit equal to the right‑hand limit. This would mislead a student into thinking no calculation is needed to find k, whereas the correct step is k-18 = -16, yielding k = 2.gpt-oss:20b: fail (misleading) 2026-10-11 — Sentence 4 incorrectly equates the limits as -16 = -16 instead of k-18 = -16, which would mislead a student into thinking the equality holds without solving for k.qwen3.6:27b-mlx: fail (error) 2026-10-11 — Line 4 claims that setting the limits equal (-16 = -16) gives k = 2, but the equation actually derived from equating the limits is k - 18 = -16, which yields k = 2. The text incorrectly states the equation used to solve for 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 2026-10-11 with SymPy 1.14.0.