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Making a piecewise function continuous

Problem 1.408 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k + 3 x^{2} & x < 0 \\ - x^{2} + x - 3 & x \ge 0 \end{cases} \) continuous for all \( \displaystyle x \).

Each piece is a polynomial, so f is continuous everywhere except possibly at x = 0. There the left-hand limit, the right-hand limit and f(c) must agree.
\[ \lim_{x \to 0^-}\left(k + 3 x^{2}\right) = k \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to 0^+}\left(- x^{2} + x - 3\right) = -3 \]
The right-hand limit, which is also f(c).✓ Proved
\[ -3 \]
Setting them equal gives k = -3.✓ Proved

Answer \( k = -3 \)

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.

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: pass
qwen3.6:27b-mlx: fail (error) — Step 4 is algebraically incorrect; equating the limits yields k = -3, but the line states -3 = -3, which is a tautology that does not solve for k.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 4 is algebraically incorrect; equating the limits yields k = -3, but the line states -3 = -3, which is a tautology that does not solve for k.
gpt-oss:20b: fail (error) 2026-10-07 — Sentence 4 incorrectly states "-3 = -3" instead of equating the left‑hand limit k to the right‑hand limit -3. The conclusion that k = -3 is not properly derived from the preceding equations.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 4 claims that setting the limits equal yields k = -3, but the equation shown is -3 = -3, which is a tautology and does not involve k. The correct equation should be k = -3.

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 2026-10-07 with SymPy 1.14.0.