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Making a piecewise function continuous

Problem 1.770 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} + 1 & x < 3 \\ 2 x^{2} + 3 x - 2 & x \ge 3 \end{cases} \) 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.
Reviewed
\[ \lim_{x \to 3^-}\left(k x^{2} + 1\right) = 9 k + 1 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to 3^+}\left(2 x^{2} + 3 x - 2\right) = 25 \]
The right-hand limit, which is also f(c).✓ Proved
\[ \left. 9 k + 1 \right|_{\substack{ k=\frac{8}{3} }} = 25 \]
Setting them equal gives k = 8/3.✓ Proved

Answer \( k = \frac{8}{3} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
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: pass 2026-10-25
qwen3.6:27b-mlx: pass 2026-10-24 — The solution correctly identifies the condition for continuity at the boundary point and performs the algebraic verification accurately.
gpt-oss:20b: fail (style) 2026-10-24 — The solution skips the explicit algebraic step of solving 9k+1=25 for k; it only verifies that k=8/3 satisfies the equation, which could mislead a student into thinking the solution was derived rather than merely checked.
qwen3.6:27b-mlx: pass 2026-10-24 — The solution correctly identifies the condition for continuity at the boundary point and performs the algebraic steps accurately.
gpt-oss:20b: pass 2026-10-24

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.