Making a piecewise function continuous
Problem 1.553 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k - 3 x^{2} & x < -1 \\ 3 x^{2} - x + 3 & x \ge -1 \end{cases} \) 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
- \[ \lim_{x \to -1^+}\left(3 x^{2} - x + 3\right) = 7 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 7 \]Setting them equal gives k = 10.✓ Proved
Answer \( k = 10 \)
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) — The solution incorrectly equates the right‑hand limit to itself (7 = 7) instead of setting the left‑hand limit k‑3 equal to the right‑hand limit 7, which would yield k = 10. This misstatement misleads the reader about the correct algebraic step.qwen3.6:27b-mlx: fail (error) — The solution claims that setting the limits equal gives k = 10, but the equation derived is k - 3 = 7, which implies k = 10 is incorrect (it should be k = 10? No, k - 3 = 7 => k = 10. Wait. Let me re-calculate. Left limit: k - 3(-1)^2 = k - 3. Right limit: 3(-1)^2 - (-1) + 3 = 3 + 1 + 3 = 7. So k - 3 = 7 => k = 10. The math is actually correct. Why did I think it was wrong? Let me re-read the prompt. 'Stated answer: k = 10'. The solution says 'Setting them equal gives k = 10'. The equation shown is '7 = 7'. This is the error. The equation shown in line 4 is '7 = 7', which does not contain k and does not lead to k = 10. The text says 'Setting them equal gives k = 10', but the equation displayed is just 7=7, which is a tautology and ignores the left-hand side expression involving k. The equation should have been k - 3 = 7 or k = 10. The displayed equation '7 = 7' is wrong/misleading because it suggests the left limit was already evaluated to 7 without k, or it's just a nonsensical step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution claims that setting the limits equal gives k = 10, but the equation derived is k - 3 = 7, which implies k = 10 is incorrect (it should be k = 10? No, k - 3 = 7 => k = 10. Wait. Let me re-calculate. Left limit: k - 3(-1)^2 = k - 3. Right limit: 3(-1)^2 - (-1) + 3 = 3 + 1 + 3 = 7. So k - 3 = 7 => k = 10. The math is actually correct. Why did I think it was wrong? Let me re-read the prompt. 'Stated answer: k = 10'. The solution says 'Setting them equal gives k = 10'. The equation shown is '7 = 7'. This is the error. The equation shown in line 4 is '7 = 7', which does not contain k and does not lead to k = 10. The text says 'Setting them equal gives k = 10', but the equation displayed is just 7=7, which is a tautology and ignores the left-hand side expression involving k. The equation should have been k - 3 = 7 or k = 10. The displayed equation '7 = 7' is wrong/misleading because it suggests the left limit was already evaluated to 7 without k, or it's just a nonsensical step.gpt-oss:20b: fail (error) 2026-10-11 — The solution incorrectly equates the right‑hand limit to itself (7 = 7) instead of setting the left‑hand limit k‑3 equal to the right‑hand limit 7, which would yield k = 10. This misstatement misleads the reader about the correct algebraic step.gpt-oss:20b: fail (error) 2026-10-11 — Sentence 4 incorrectly equates the limits as 7 = 7 instead of setting k - 3 = 7, which leads to the correct value k = 10. The conclusion is not derived properly from the previous equations.qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution claims that setting the limits equal (k - 3 = 7) results in k = 10, but the equation shown is '7 = 7', which is incorrect and does not lead to the stated answer. The correct equation should be k - 3 = 7, yielding k = 10.
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 2026-10-11 with SymPy 1.14.0.