Making a piecewise function continuous
Problem 1.294 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} + 4 & x < 3 \\ 3 x + 1 & x \ge 3 \end{cases} \) continuous for all \( \displaystyle x \).
- Each piece is a polynomial, so f is continuous everywhere except possibly at x = 3. There the left-hand limit, the right-hand limit and f(c) must agree.
- \[ \lim_{x \to 3^-}\left(k x^{2} + 4\right) = 9 k + 4 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 3^+}\left(3 x + 1\right) = 10 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 10 \]Setting them equal gives k = 2/3.✓ Proved
Answer \( k = \frac{2}{3} \)
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 (misleading) — Sentence 4 incorrectly states "10 = 10" instead of equating the left‑hand limit 9k+4 to the right‑hand limit 10, which misleads a student about the algebraic step needed to solve for k.qwen3.6:27b-mlx: fail (error) — The solution fails to actually equate the left-hand limit (9k + 4) with the right-hand limit (10). Instead, it presents a tautology '10 = 10' and claims this yields k = 2/3, omitting the necessary algebraic step 9k + 4 = 10.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to actually equate the left-hand limit (9k + 4) with the right-hand limit (10). Instead, it presents a tautology '10 = 10' and claims this yields k = 2/3, omitting the necessary algebraic step 9k + 4 = 10.gpt-oss:20b: fail (misleading) 2026-10-05 — Sentence 4 incorrectly states "10 = 10" instead of equating the left‑hand limit 9k+4 to the right‑hand limit 10, which misleads a student about the algebraic step needed to solve for k.qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 4 claims that setting the limits equal yields k = 2/3, but the equation shown is '10 = 10', which is a tautology and does not involve k. The correct equation should be 9k + 4 = 10.gpt-oss:20b: fail (error) 2026-10-05 — The solution incorrectly sets the left and right limits equal. It writes "10 = 10" instead of solving 9k+4 = 10, which leads to the correct value k = 2/3. The conclusion is not justified by the preceding equations.
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 2026-10-05 with SymPy 1.14.0.