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Making a piecewise function continuous

Problem 1.489 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} + 4 & x < -2 \\ - x^{2} - 2 x + 3 & x \ge -2 \end{cases} \) 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.
\[ \lim_{x \to -2^-}\left(k x^{2} + 4\right) = 4 k + 4 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to -2^+}\left(- x^{2} - 2 x + 3\right) = 3 \]
The right-hand limit, which is also f(c).✓ Proved
\[ 3 \]
Setting them equal gives k = -1/4.✓ Proved

Answer \( k = - \frac{1}{4} \)

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.

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 claims that setting the limits equal results in '3 = 3', which is incorrect; it should be '4k + 4 = 3'. The equation shown does not involve k and does not lead to the stated answer.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 claims that setting the limits equal results in '3 = 3', which is incorrect; it should be '4k + 4 = 3'. The equation shown does not involve k and does not lead to the stated answer.
gpt-oss:20b: fail (misleading) 2026-10-09 — The solution skips the crucial step of equating the left‐hand limit 4k+4 with the right‐hand limit 3; it simply states 3=3 and then gives k=-1/4, which could mislead a student into thinking the equality was already satisfied.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution claims that setting the limits equal (4k + 4 = 3) yields k = -1/4, but the correct solution is k = -1/4 only if 4k + 4 = 3 implies 4k = -1, which is correct. Wait, let me re-read. 4k + 4 = 3 => 4k = -1 => k = -1/4. The math is actually correct. Let me re-read the prompt's provided solution text carefully. Line 4 says '3 = 3 -- Setting them equal gives k = -1/4'. This is a massive logical error in the text of the sentence. It states '3 = 3' as the equation resulting from setting them equal, which is false (the equation is 4k+4=3). It then claims this false equation gives k=-1/4. The sentence is nonsensical and misleading.

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 2026-10-09 with SymPy 1.14.0.