Making a piecewise function continuous
Problem 1.206 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k + 3 x^{2} & x < 1 \\ - x^{2} - 3 x & x \ge 1 \end{cases} \) 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.
- \[ \lim_{x \to 1^-}\left(k + 3 x^{2}\right) = k + 3 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 1^+}\left(- x^{2} - 3 x\right) = -4 \]The right-hand limit, which is also f(c).✓ Proved
- \[ -4 \]Setting them equal gives k = -7.✓ Proved
Answer \( k = -7 \)
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 equality. It should equate the left‑hand limit k+3 to the right‑hand limit -4, yielding k = -7. The line "-4 = -4" does not reflect this step and misleads the reader.qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal (-4 = -4) gives k = -7, but the equation k + 3 = -4 is required to solve for k. The displayed equation is a tautology that does not involve k.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 4 claims that setting the limits equal (-4 = -4) gives k = -7, but the equation k + 3 = -4 is required to solve for k. The displayed equation is a tautology that does not involve k.gpt-oss:20b: fail (error) 2026-10-04 — Sentence 4 incorrectly sets the equality. It should equate the left‑hand limit k+3 to the right‑hand limit -4, yielding k = -7. The line "-4 = -4" does not reflect this step and misleads the reader.qwen3.6:27b-mlx: fail (error) 2026-10-04 — Line 4 claims that setting the limits equal results in -4 = -4, which is a tautology and does not solve for k. The correct equation is k + 3 = -4, which yields k = -7.gpt-oss:20b: fail (misleading) 2026-10-04 — The solution incorrectly states that the left and right limits are equal by writing "-4 = -4"; it should set k+3 = -4 to solve for k = -7.
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 2026-10-04 with SymPy 1.14.0.