Making a piecewise function continuous
Problem 1.549 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} - 3 & x < 2 \\ 2 - 4 x & x \ge 2 \end{cases} \) 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.
- \[ \lim_{x \to 2^-}\left(k x^{2} - 3\right) = 4 k - 3 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 2^+}\left(2 - 4 x\right) = -6 \]The right-hand limit, which is also f(c).✓ Proved
- \[ -6 \]Setting them equal gives k = -3/4.✓ Proved
Answer \( k = - \frac{3}{4} \)
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 (misleading) — The final sentence claims that setting the limits equal gives k = -3/4, but the equation shown is -6 = -6, which does not reflect the necessary equation 4k-3 = -6. This misleads the reader about how the value of k is obtained.qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal results in the equation '-6 = -6', which is incorrect. The correct equation is '4k - 3 = -6'. This is a fundamental algebraic error in the setup of the continuity condition.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — Step 4 claims that setting the limits equal results in the equation '-6 = -6', which is incorrect. The correct equation is '4k - 3 = -6'. This is a fundamental algebraic error in the setup of the continuity condition.gpt-oss:20b: fail (misleading) 2026-10-11 — The final sentence claims that setting the limits equal gives k = -3/4, but the equation shown is -6 = -6, which does not reflect the necessary equation 4k-3 = -6. This misleads the reader about how the value of k is obtained.gpt-oss:20b: fail (error) 2026-10-11 — Sentence 4 incorrectly sets the limits equal to each other as -6 = -6 instead of solving 4k-3 = -6 for k. The conclusion that k = -3/4 is not derived from the given equations.qwen3.6:27b-mlx: fail (error) 2026-10-11 — Line 4 claims that setting the limits equal results in '-6 = -6', which is a tautology and does not involve k. The correct equation should be '4k - 3 = -6'. This is a critical algebraic error in the setup of the final step.
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 2026-10-11 with SymPy 1.14.0.