Making a piecewise function continuous
Problem 1.492 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} - 5 & x < 2 \\ 3 x + 3 & x \ge 2 \end{cases} \) 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.
- \[ \lim_{x \to 2^-}\left(k x^{2} - 5\right) = 4 k - 5 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 2^+}\left(3 x + 3\right) = 9 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 9 \]Setting them equal gives k = 7/2.✓ Proved
Answer \( k = \frac{7}{2} \)
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 states that the right‑hand limit equals 9 and then writes "9 = 9" as the equation to solve. It should set the left‑hand limit 4k‑5 equal to the right‑hand limit 9, i.e., 4k‑5 = 9, to find k = 7/2. The current statement misrepresents the algebraic step and could mislead a student.qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal results in 9 = 9, which is incorrect. The correct equation is 4k - 5 = 9. This is a fundamental algebraic error in the solution process.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — The solution incorrectly states that the right‑hand limit equals 9 and then writes "9 = 9" as the equation to solve. It should set the left‑hand limit 4k‑5 equal to the right‑hand limit 9, i.e., 4k‑5 = 9, to find k = 7/2. The current statement misrepresents the algebraic step and could mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 claims that setting the limits equal results in 9 = 9, which is incorrect. The correct equation is 4k - 5 = 9. This is a fundamental algebraic error in the solution process.gpt-oss:20b: fail (misleading) 2026-10-09 — The solution incorrectly states that the left- and right-hand limits are both 9; it should set 4k-5 = 9 to solve for k.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Line 4 claims that setting the limits equal (4k - 5 = 9) results in the equation 9 = 9, which is mathematically incorrect and fails to show the actual step of solving for k.
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.