Home›Calculus 1›Making a piecewise function continuous›Problem 1.555

Making a piecewise function continuous

Problem 1.555 · medium

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

Each piece is a polynomial, so f is continuous everywhere except possibly at x = 3. There the left-hand limit, the right-hand limit and f(c) must agree.
Reviewed
\[ \lim_{x \to 3^-}\left(k + x^{2}\right) = k + 9 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to 3^+}\left(3 x^{2} + 2 x - 1\right) = 32 \]
The right-hand limit, which is also f(c).✓ Proved
\[ 32 \]
Setting them equal gives k = 23.✓ Proved

Answer \( k = 23 \)

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. Each piece is a polynomial and so is continuous on its own open region, which leaves only x = 3. Continuity there does require the left limit, the right limit and f(3) to agree.
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 equates the two limits as 32 = 32 instead of setting k + 9 = 32 to solve for k. This would mislead a student into thinking no value of k is needed or that the equation is already satisfied.
qwen3.6:27b-mlx: fail (error) — The solution claims that setting the limits equal gives k = 23, but the equation shown is 32 = 32, which is a tautology and does not involve k. The correct step should equate k + 9 to 32.

Senior review claude-sonnet-5-5, 2026-10-11: fail — Lines 1-3 are correct: k + 9 is the left limit and 32 is the right limit and f(3). Line 4 should read k + 9 = 32, then k = 23. As written, a student sees a trivial identity and a claim that it 'gives k = 23', with the solving step missing.

qwen3.6:27b-mlx: uphold — Line 4 reads '32 = 32', a tautology with no k in it. The equation k + 9 = 32 and its solution k = 23 never appear, so the stated answer is not derived on the page.

Every verdict on record (5)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution claims that setting the limits equal gives k = 23, but the equation shown is 32 = 32, which is a tautology and does not involve k. The correct step should equate k + 9 to 32.
gpt-oss:20b: fail (misleading) 2026-10-11 — The solution incorrectly equates the two limits as 32 = 32 instead of setting k + 9 = 32 to solve for k. This would mislead a student into thinking no value of k is needed or that the equation is already satisfied.
claude-sonnet-5-5: fail (misleading) 2026-10-11 — Lines 1-3 are correct: k + 9 is the left limit and 32 is the right limit and f(3). Line 4 should read k + 9 = 32, then k = 23. As written, a student sees a trivial identity and a claim that it 'gives k = 23', with the solving step missing.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The equation in step 4 is '32 = 32', which is a tautology and does not involve k. It fails to set the left-hand limit (k + 9) equal to the right-hand limit (32), making the derivation of k = 23 logically unsupported.

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.