Making a piecewise function continuous
Problem 1.402 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} + 1 & x < 2 \\ - x^{2} & x \ge 2 \end{cases} \) 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
- \[ \lim_{x \to 2^+}\left(- x^{2}\right) = -4 \]The right-hand limit, which is also f(c).✓ Proved
- \[ -4 \]Setting them equal gives k = -5/4.✓ Proved
Answer \( k = - \frac{5}{4} \)
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) — The solution incorrectly equates the limits as -4 = -4 instead of setting 4k+1 = -4 to solve for k. This misleads the reader and gives an incorrect reasoning step.qwen3.6:27b-mlx: fail (error) — Line 4 claims that setting the limits equal results in '-4 = -4', which is a tautology and does not involve k. The correct equation should be '4k + 1 = -4'.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — The solution incorrectly equates the limits as -4 = -4 instead of setting 4k+1 = -4 to solve for k. This misleads the reader and gives an incorrect reasoning step.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Line 4 claims that setting the limits equal results in '-4 = -4', which is a tautology and does not involve k. The correct equation should be '4k + 1 = -4'.gpt-oss:20b: fail (error) 2026-10-07 — The solution incorrectly states that the left- and right-hand limits are equal as "-4 = -4", which is a tautology. The correct step is to set 4k+1 = -4 and solve for k, yielding k = -5/4. The erroneous equality misleads the reader about how the value of k is obtained.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 4 claims that setting the limits equal results in -4 = -4, which is a tautology and does not involve k. The correct equation is 4k + 1 = -4.
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 2026-10-07 with SymPy 1.14.0.