Making a piecewise function continuous
Problem 1.547 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} - 3 & x < 3 \\ 4 x & x \ge 3 \end{cases} \) 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.
- \[ \lim_{x \to 3^-}\left(k x^{2} - 3\right) = 9 k - 3 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 3^+}\left(4 x\right) = 12 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 12 \]Setting them equal gives k = 5/3.✓ Proved
Answer \( k = \frac{5}{3} \)
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) — Step 4 incorrectly states "12 = 12" instead of equating the left‑hand limit 9k‑3 to the right‑hand limit 12 and solving for k. This could mislead a student into thinking the equality holds automatically, hiding the necessary algebraic step.qwen3.6:27b-mlx: fail (error) — The solution fails to actually set the left-hand limit equal to the right-hand limit. It states the condition but then shows a tautology (12 = 12) instead of the equation 9k - 3 = 12, omitting the crucial step of solving for k.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to actually set the left-hand limit equal to the right-hand limit. It states the condition but then shows a tautology (12 = 12) instead of the equation 9k - 3 = 12, omitting the crucial step of solving for k.gpt-oss:20b: fail (misleading) 2026-10-11 — Step 4 incorrectly states "12 = 12" instead of equating the left‑hand limit 9k‑3 to the right‑hand limit 12 and solving for k. This could mislead a student into thinking the equality holds automatically, hiding the necessary algebraic step.gpt-oss:20b: fail (misleading) 2026-10-11 — The final sentence incorrectly states "12 = 12" as the result of equating the limits, which does not show the calculation leading to k = 5/3. The student would be misled into thinking the equality holds automatically without solving for k.qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to actually equate the left-hand limit (9k - 3) with the right-hand limit (12). Instead, it presents the tautology '12 = 12' and claims this yields k = 5/3, omitting the crucial algebraic step 9k - 3 = 12.
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.