Making a piecewise function continuous
Problem 1.202 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k + 3 x^{2} & x < 2 \\ 4 x + 4 & x \ge 2 \end{cases} \) 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.
- \[ \lim_{x \to 2^-}\left(k + 3 x^{2}\right) = k + 12 \]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 = 0.✓ Proved
Answer \( k = 0 \)
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 sets the left‐hand limit equal to the right‐hand limit as 12 = 12, omitting the term k + 12. The correct equation is k + 12 = 12, which yields k = 0.qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal (k + 12 = 12) gives k = 0, but the equation shown is '12 = 12', which is a tautology and does not involve k. The step fails to show the actual equation k + 12 = 12.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 4 claims that setting the limits equal (k + 12 = 12) gives k = 0, but the equation shown is '12 = 12', which is a tautology and does not involve k. The step fails to show the actual equation k + 12 = 12.gpt-oss:20b: fail (error) 2026-10-04 — The solution incorrectly sets the left‐hand limit equal to the right‐hand limit as 12 = 12, omitting the term k + 12. The correct equation is k + 12 = 12, which yields k = 0.qwen3.6:27b-mlx: fail (error) 2026-10-04 — Line 4 claims that setting the limits equal gives k=0, but the equation shown is 12=12, which ignores the variable k entirely. The correct equation should be k + 12 = 12.gpt-oss:20b: fail (error) 2026-10-04 — Sentence 4 incorrectly equates the limits as 12 = 12, omitting the term k + 12. The correct equation should be k + 12 = 12, which yields k = 0.
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.