Making a piecewise function continuous
Problem 1.293 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k - 3 x^{2} & x < 3 \\ - 3 x - 2 & x \ge 3 \end{cases} \) 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.
- \[ \lim_{x \to 3^-}\left(k - 3 x^{2}\right) = k - 27 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 3^+}\left(- 3 x - 2\right) = -11 \]The right-hand limit, which is also f(c).✓ Proved
- \[ -11 \]Setting them equal gives k = 16.✓ Proved
Answer \( k = 16 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in 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: passqwen3.6:27b-mlx: fail (error) — The solution sets up the correct limits but fails to equate them to solve for k. Line 4 shows '-11 = -11', which is a tautology derived from the right-hand limit alone, ignoring the left-hand limit 'k - 27'. The step to set 'k - 27 = -11' is missing.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution sets up the correct limits but fails to equate them to solve for k. Line 4 shows '-11 = -11', which is a tautology derived from the right-hand limit alone, ignoring the left-hand limit 'k - 27'. The step to set 'k - 27 = -11' is missing.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to actually solve for k. It states the left-hand limit is k - 27 and the right-hand limit is -11, but the final equation shown is '-11 = -11' instead of 'k - 27 = -11'. This skips the crucial algebraic step and presents a tautology rather than the condition for continuity.gpt-oss:20b: fail (error) 2026-10-05 — Sentence 4 incorrectly states the equality; it should equate k-27 with -11 to solve for k, not simply write -11 = -11.
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 2026-10-05 with SymPy 1.14.0.