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Making a piecewise function continuous

Problem 1.406 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} + 3 & x < -1 \\ 3 x^{2} + x + 1 & x \ge -1 \end{cases} \) 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.
\[ \lim_{x \to -1^-}\left(k x^{2} + 3\right) = k + 3 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to -1^+}\left(3 x^{2} + x + 1\right) = 3 \]
The right-hand limit, which is also f(c).✓ Proved
\[ 3 \]
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) — Step 4 claims that setting the limits equal (k + 3 = 3) yields k = 0, but the equation shown is '3 = 3', which is a tautology and does not solve for k. The correct equation should be k + 3 = 3.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 4 claims that setting the limits equal (k + 3 = 3) yields k = 0, but the equation shown is '3 = 3', which is a tautology and does not solve for k. The correct equation should be k + 3 = 3.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution claims that setting the left-hand limit (k + 3) equal to the right-hand limit (3) yields k = 0. However, k + 3 = 3 implies k = 0 is incorrect; it implies k = 0 only if the equation was k + 3 = 3, which gives k=0. Wait, k+3=3 => k=0. Let me re-read. Left limit: k(-1)^2 + 3 = k + 3. Right limit: 3(-1)^2 + (-1) + 1 = 3 - 1 + 1 = 3. Equation: k + 3 = 3 => k = 0. The algebra is actually correct. k=0 is the correct answer. Let's look closer at the sentences. Sentence 1: "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." This is correct. Sentence 4: "Setting them equal gives k = 0." The equation shown is `3 = 3`. This is the problem. The equation displayed in line 4 is `3 = 3`, which is a tautology and does not show the step `k + 3 = 3`. The text says "Setting them equal gives k=0", but the equation line shows `3=3`. This is a mismatch between the equation line and the text, or rather, the equation line is missing the variable k entirely. It seems the solver substituted k=0 into the left limit *before* solving, or just wrote the right limit equals the right limit. Actually, looking at line 4: `[equation, unchecked] 3 = 3 -- Setting them equal gives k = 0.` This is logically flawed presentation. You set `k+3 = 3`. You do not set `3=3`. The equation line should have been `k + 3 = 3`. By writing `3=3`, it implies the left limit was already evaluated to 3, which assumes k=0, creating a circular argument or a typo. A student would be confused why `3=3` leads to `k=0`. It fails to show the dependency on k.
gpt-oss:20b: inconclusive 2026-10-07 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly sets the left‑hand limit equal to the right‑hand limit. It writes \"3 = 3\" instead of equating \(k+3\) to 3, which would correct

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.