Home›Calculus 1›Making a piecewise function continuous›Problem 1.409

Making a piecewise function continuous

Problem 1.409 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k - 2 x^{2} & x < -2 \\ 2 x^{2} + 2 x & x \ge -2 \end{cases} \) 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.
Reviewed
\[ \lim_{x \to -2^-}\left(k - 2 x^{2}\right) = k - 8 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to -2^+}\left(2 x^{2} + 2 x\right) = 4 \]
The right-hand limit, which is also f(c).✓ Proved
\[ 4 \]
Setting them equal gives k = 12.✓ Proved

Answer \( k = 12 \)

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: This is a sentence, not an algebraic equivalence. Polynomials are continuous, so only x = -2 can be a point of discontinuity. Continuity there requires the left limit, right limit and f(-2) to agree, which is the correct criterion.
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 (error) — Sentence 4 incorrectly states 4 = 4 instead of setting the left‐hand limit k−8 equal to the right‐hand limit 4, which would yield k = 12. The conclusion is not properly derived from the previous lines.
qwen3.6:27b-mlx: fail (error) — The solution claims that setting the limits equal yields k = 12, but the equation derived from the limits is k - 8 = 4, which implies k = 12. However, line 4 states '4 = 4', which is a tautology and does not represent the equation k - 8 = 4. The logic connecting the limits to the value of k is missing or incorrectly stated in line 4.

Senior review claude-sonnet-5-5, 2026-10-07: fail — The answer k = 12 is correct (12 - 8 = 4, and both one-sided limits equal f(-2) = 4). Line 4 should read k - 8 = 4, then k = 12. As written, the equation that determines k is skipped and a tautology is shown in its place.

qwen3.6:27b-mlx: uphold — The final line '4 = 4' is true but vacuous: the equation k - 8 = 4 that actually determines k never appears, so the caption 'gives k = 12' is not supported by what is displayed. The math is not wrong, only the severity 'error' is overstated.

Every verdict on record (5)

gpt-oss:20b: fail (error) 2026-10-07 — Sentence 4 incorrectly states 4 = 4 instead of setting the left‐hand limit k−8 equal to the right‐hand limit 4, which would yield k = 12. The conclusion is not properly derived from the previous lines.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution claims that setting the limits equal yields k = 12, but the equation derived from the limits is k - 8 = 4, which implies k = 12. However, line 4 states '4 = 4', which is a tautology and does not represent the equation k - 8 = 4. The logic connecting the limits to the value of k is missing or incorrectly stated in line 4.
claude-sonnet-5-5: fail (misleading) 2026-10-07 — The answer k = 12 is correct (12 - 8 = 4, and both one-sided limits equal f(-2) = 4). Line 4 should read k - 8 = 4, then k = 12. As written, the equation that determines k is skipped and a tautology is shown in its place.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution claims that setting the limits equal gives k = 12, but the equation shown is '4 = 4', which is independent of k. The correct equation should be k - 8 = 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.