Making a piecewise function continuous
Problem 1.407 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k - x^{2} & x < 3 \\ 2 x^{2} - 2 x - 3 & x \ge 3 \end{cases} \) 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.
- \[ \lim_{x \to 3^-}\left(k - x^{2}\right) = k - 9 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 3^+}\left(2 x^{2} - 2 x - 3\right) = 9 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 9 \]Setting them equal gives k = 18.✓ Proved
Answer \( k = 18 \)
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) — Step 4 incorrectly sets the limits equal as 9 = 9 instead of equating k‑9 to 9, which leads to the correct conclusion k = 18 but the reasoning is flawed.qwen3.6:27b-mlx: fail (error) — The solution sets the right-hand limit (9) equal to itself, ignoring the left-hand limit expression (k - 9). It fails to equate the two limits to solve for k.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Step 4 incorrectly sets the limits equal as 9 = 9 instead of equating k‑9 to 9, which leads to the correct conclusion k = 18 but the reasoning is flawed.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution sets the right-hand limit (9) equal to itself, ignoring the left-hand limit expression (k - 9). It fails to equate the two limits to solve for k.gpt-oss:20b: fail (misleading) 2026-10-07 — The solution equates the limits as 9 = 9, omitting the variable k. The correct step is k - 9 = 9, which yields k = 18. This omission could mislead a student into thinking the equality holds regardless of k.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution sets the right-hand limit (9) equal to itself (9) instead of equating the left-hand limit (k - 9) to the right-hand limit. This algebraic error fails to solve for k, despite stating the correct final answer.
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.