Home›Calculus 1›Making a piecewise function continuous›Problem 1.546

Making a piecewise function continuous

Problem 1.546 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k + 2 x^{2} & x < -1 \\ - 2 x^{2} - 3 x + 1 & x \ge -1 \end{cases} \) continuous for all \( \displaystyle x \).

Each piece is a polynomial, so f is continuous everywhere except possibly at x = -1. There the left-hand limit, the right-hand limit and f(c) must agree.
\[ \lim_{x \to -1^-}\left(k + 2 x^{2}\right) = k + 2 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to -1^+}\left(- 2 x^{2} - 3 x + 1\right) = 2 \]
The right-hand limit, which is also f(c).✓ Proved
\[ 2 \]
Setting them equal gives k = 0.✓ Proved

Answer \( k = 0 \)

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) — Line 4 claims that setting the limits equal gives k = 0, but the equation shown is '2 = 2', which is derived from the right-hand limit alone and ignores the left-hand limit (k + 2). The correct equation should be k + 2 = 2.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — Line 4 claims that setting the limits equal gives k = 0, but the equation shown is '2 = 2', which is derived from the right-hand limit alone and ignores the left-hand limit (k + 2). The correct equation should be k + 2 = 2.
gpt-oss:20b: pass 2026-10-11
gpt-oss:20b: fail (error) 2026-10-11 — The conclusion step incorrectly equates 2=2 instead of setting k+2=2, which would lead to k=0. The reasoning is otherwise sound.
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution claims that setting the limits equal gives k=0, but the equation shown is '2 = 2', which ignores the variable k entirely. The correct equation should be k + 2 = 2.

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.