Making a piecewise function continuous
Problem 1.488 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k - 3 x^{2} & x < 0 \\ 2 x^{2} - x + 2 & x \ge 0 \end{cases} \) continuous for all \( \displaystyle x \).
- Each piece is a polynomial, so f is continuous everywhere except possibly at x = 0. There the left-hand limit, the right-hand limit and f(c) must agree.
- \[ \lim_{x \to 0^-}\left(k - 3 x^{2}\right) = k \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 0^+}\left(2 x^{2} - x + 2\right) = 2 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 2 \]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 (error) — Sentence 4 incorrectly states "2 = 2" instead of "k = 2". The conclusion is correct, but the equation is wrong.qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal gives k = 2, but the previous steps established the left-hand limit as k and the right-hand limit as 2. Equating them yields k = 2, but the equation shown is '2 = 2', which is a tautology that does not involve k and fails to demonstrate the derivation of the value for k.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — Sentence 4 incorrectly states "2 = 2" instead of "k = 2". The conclusion is correct, but the equation is wrong.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 claims that setting the limits equal gives k = 2, but the previous steps established the left-hand limit as k and the right-hand limit as 2. Equating them yields k = 2, but the equation shown is '2 = 2', which is a tautology that does not involve k and fails to demonstrate the derivation of the value for k.gpt-oss:20b: fail (misleading) 2026-10-09 — The final sentence claims that setting the limits equal gives k=2, yet the equation shown is 2=2, which does not involve k. This misleads the reader into thinking the equality was verified, but the actual step equating k to 2 is missing.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 claims that setting the limits equal gives k = 2, but the equation derived is k = 2. The text '2 = 2' is incorrect; it should be 'k = 2'. The left-hand limit is k, and the right-hand limit is 2, so equating them yields k = 2, not 2 = 2.
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.