Making a piecewise function continuous
Problem 1.207 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k - 2 x^{2} & x < 2 \\ x + 4 & x \ge 2 \end{cases} \) continuous for all \( \displaystyle x \).
- Each piece is a polynomial, so f is continuous everywhere except possibly at x = 2. There the left-hand limit, the right-hand limit and f(c) must agree.
- \[ \lim_{x \to 2^-}\left(k - 2 x^{2}\right) = k - 8 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 2^+}\left(x + 4\right) = 6 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 6 \]Setting them equal gives k = 14.✓ Proved
Answer \( k = 14 \)
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 sets 6=6 instead of equating the left‑hand limit k-8 to the right‑hand limit 6, which would yield k=14.qwen3.6:27b-mlx: fail (error) — Line 4 claims that setting the limits equal gives k = 14, but the equation shown is '6 = 6', which is independent of k. The correct step should equate the left-hand limit (k - 8) to the right-hand limit (6), i.e., k - 8 = 6.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — Line 4 claims that setting the limits equal gives k = 14, but the equation shown is '6 = 6', which is independent of k. The correct step should equate the left-hand limit (k - 8) to the right-hand limit (6), i.e., k - 8 = 6.gpt-oss:20b: fail (error) 2026-10-04 — Sentence 4 incorrectly sets 6=6 instead of equating the left‑hand limit k-8 to the right‑hand limit 6, which would yield k=14.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to actually equate the left-hand limit (k - 8) to the right-hand limit (6). Instead, it presents '6 = 6' and claims this yields k = 14, which is a logical non-sequitur and algebraically incorrect.gpt-oss:20b: fail (error) 2026-10-04 — Step 4 incorrectly sets the limits equal as 6 = 6 instead of solving k - 8 = 6 to find k = 14.
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 2026-10-04 with SymPy 1.14.0.