Making a piecewise function continuous
Problem 1.495 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} - 1 & x < 2 \\ 4 x + 4 & x \ge 2 \end{cases} \) 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.
- \[ \lim_{x \to 2^-}\left(k x^{2} - 1\right) = 4 k - 1 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 2^+}\left(4 x + 4\right) = 12 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 12 \]Setting them equal gives k = 13/4.✓ Proved
Answer \( k = \frac{13}{4} \)
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 left‑hand limit equal to the right‑hand limit. It should solve 4k‑1 = 12, yielding k = 13/4, not simply state 12 = 12.qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal results in '12 = 12', which is incorrect; it should be '4k - 1 = 12'. This algebraic error obscures the actual calculation required to find k.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — Step 4 incorrectly sets the left‑hand limit equal to the right‑hand limit. It should solve 4k‑1 = 12, yielding k = 13/4, not simply state 12 = 12.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 claims that setting the limits equal results in '12 = 12', which is incorrect; it should be '4k - 1 = 12'. This algebraic error obscures the actual calculation required to find k.gpt-oss:20b: fail (error) 2026-10-09 — Sentence 4 incorrectly states 12 = 12; it should set 4k-1 = 12 to solve for k, yielding k = 13/4.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Line 4 is algebraically incorrect; it states '12 = 12' but claims this implies k = 13/4. The correct equation derived from equating the limits is 4k - 1 = 12.
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 2026-10-09 with SymPy 1.14.0.