Making a piecewise function continuous
Problem 1.765 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k + 2 x^{2} & x < 3 \\ - 3 x^{2} - 2 x - 1 & x \ge 3 \end{cases} \) 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.Reviewed
- \[ \lim_{x \to 3^-}\left(k + 2 x^{2}\right) = k + 18 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 3^+}\left(- 3 x^{2} - 2 x - 1\right) = -34 \]The right-hand limit, which is also f(c).✓ Proved
- \[ \left. k + 18 \right|_{\substack{ k=-52 }} = -34 \]Setting them equal gives k = -52.✓ Proved
Answer \( k = -52 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the condition for continuity at the boundary point and performs the algebraic steps accurately.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly identifies the condition for continuity at the boundary point and performs the algebraic steps accurately.gpt-oss:20b: pass 2026-10-25qwen3.6:27b-mlx: pass 2026-10-24 — The solution correctly identifies the condition for continuity at the boundary point and performs the algebraic verification accurately.gpt-oss:20b: pass 2026-10-24qwen3.6:27b-mlx: pass 2026-10-24 — The solution correctly identifies the condition for continuity at the boundary point and performs the algebraic steps accurately.gpt-oss:20b: pass 2026-10-24
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 2026-10-25 with SymPy 1.14.0.