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Making a piecewise function continuous

Problem 1.403 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x + 1 & x < 1 \\ - 2 x^{2} - 3 x + 1 & x \ge 1 \end{cases} \) 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.
Reviewed
\[ \lim_{x \to 1^-}\left(k x + 1\right) = k + 1 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to 1^+}\left(- 2 x^{2} - 3 x + 1\right) = -4 \]
The right-hand limit, which is also f(c).✓ Proved
\[ -4 \]
Setting them equal gives k = -5.✓ Proved

Answer \( k = -5 \)

Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: A sentence, not an algebraic equivalence. Both pieces are polynomials, hence continuous on their open domains, so only x = 1 needs checking. Continuity there requires the left limit, right limit and f(1) to agree, which is the standard definition.
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 incorrectly states that the right‑hand limit equals the left‑hand limit as "-4 = -4". It omits the equation k+1 = -4 that must be solved to find k = -5. This misleads a student into thinking no calculation is needed.
qwen3.6:27b-mlx: fail (error) — The final equation '-4 = -4' is incorrect; it should be 'k + 1 = -4'. The text claims this incorrect equation gives k = -5, but the algebraic step is missing and the displayed equation is false.

Senior review claude-sonnet-5-5, 2026-10-07: fail — The answer k = -5 is correct (k + 1 = -4), but line 4 displays -4 = -4 instead of k + 1 = -4 and then k = -5. A student would not see how k is determined. Lines 1-3 are fine.

qwen3.6:27b-mlx: uphold — The displayed line -4 = -4 is a tautology with no k, so it does not show the equation k + 1 = -4 that its caption says yields k = -5; the solving step is never actually written.

Every verdict on record (5)

gpt-oss:20b: fail (misleading) 2026-10-07 — The solution incorrectly states that the right‑hand limit equals the left‑hand limit as "-4 = -4". It omits the equation k+1 = -4 that must be solved to find k = -5. This misleads a student into thinking no calculation is needed.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The final equation '-4 = -4' is incorrect; it should be 'k + 1 = -4'. The text claims this incorrect equation gives k = -5, but the algebraic step is missing and the displayed equation is false.
claude-sonnet-5-5: fail (misleading) 2026-10-07 — The answer k = -5 is correct (k + 1 = -4), but line 4 displays -4 = -4 instead of k + 1 = -4 and then k = -5. A student would not see how k is determined. Lines 1-3 are fine.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — Line 4 claims that setting the limits equal yields k = -5, but the equation shown is '-4 = -4', which is a tautology and does not involve k. The correct equation should be 'k + 1 = -4'.

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.