Making a piecewise function continuous
Problem 1.768 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} - 2 & x < 2 \\ 2 x + 1 & x \ge 2 \end{cases} \) 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.
- \[ \lim_{x \to 2^-}\left(k x^{2} - 2\right) = 4 k - 2 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 2^+}\left(2 x + 1\right) = 5 \]The right-hand limit, which is also f(c).✓ Proved
- \[ \left. 4 k - 2 \right|_{\substack{ k=\frac{7}{4} }} = 5 \]Setting them equal gives k = 7/4.✓ Proved
Answer \( k = \frac{7}{4} \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
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 solution skips the algebraic step of solving 4k-2=5 for k; it merely plugs in k=7/4 to verify, which could mislead a student into thinking the value was derived rather than solved.qwen3.6:27b-mlx: pass — The solution correctly identifies the condition for continuity at the piecewise boundary and solves for k accurately.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly identifies the condition for continuity at the piecewise boundary and solves for k accurately.gpt-oss:20b: fail (misleading) 2026-10-25 — The solution skips the algebraic step of solving 4k-2=5 for k; it merely plugs in k=7/4 to verify, which could mislead a student into thinking the value was derived rather than solved.qwen3.6:27b-mlx: pass 2026-10-24 — The solution correctly identifies the condition for continuity at the boundary and performs the algebraic steps accurately.gpt-oss:20b: fail (misleading) 2026-10-24 — Step 4 claims to solve for k by substituting k=7/4, but it only verifies the equality; the actual solution requires solving 4k-2=5 to obtain k=7/4.qwen3.6:27b-mlx: pass 2026-10-24 — The solution correctly identifies the condition for continuity at the boundary point and solves for k accurately.gpt-oss:20b: fail (misleading) 2026-10-24 — The solution skips the crucial step of solving 4k-2=5 for k; it merely substitutes k=7/4 and checks the equality, which does not justify that this is the unique value that makes the function continuous.
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 2026-10-25 with SymPy 1.14.0.