Antiderivatives with initial conditions
Problem 3.375 · hard
Solve \( \displaystyle f''(x) = - 4 x^{2} - 3 \cos{\left(x \right)} \) with \( \displaystyle f'(1) = -6 \) and \( \displaystyle f(1) = -2 \).
- \[ \frac{d}{d x} \left(- \frac{4 x^{3}}{3} - 3 \sin{\left(x \right)}\right) = - 4 x^{2} - 3 \cos{\left(x \right)} \]An antiderivative of f″; f′ is this plus a constant C₁.✓ Proved
- \[ - \left. - \frac{4 x^{3}}{3} - 3 \sin{\left(x \right)} \right|_{\substack{ x=1 }} - 6 = - \frac{14}{3} + 3 \sin{\left(1 \right)} \]f′(1) = -6 fixes C₁.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{4}}{3} - \frac{14 x}{3} + 3 x \sin{\left(1 \right)} + 3 \cos{\left(x \right)}\right) = - \frac{4 x^{3}}{3} - 3 \sin{\left(x \right)} - \frac{14}{3} + 3 \sin{\left(1 \right)} \]Antidifferentiate f′; add a constant C₀.✓ Proved
- \[ - \left. - \frac{14 x}{3} + 3 x \sin{\left(1 \right)} + \frac{\left(-1\right) x^{4}}{3} + 3 \cos{\left(x \right)} \right|_{\substack{ x=1 }} - 2 = - 3 \sin{\left(1 \right)} - 3 \cos{\left(1 \right)} + 3 \]f(1) = -2 fixes C₀.✓ Proved
- \[ - \frac{14 x}{3} + 3 x \sin{\left(1 \right)} + \frac{\left(-1\right) x^{4}}{3} + 3 \cos{\left(x \right)} - 3 \sin{\left(1 \right)} - 3 \cos{\left(1 \right)} + 3 = - \frac{x^{4}}{3} - \frac{14 x}{3} + 3 x \sin{\left(1 \right)} + 3 \cos{\left(x \right)} - 3 \sin{\left(1 \right)} - 3 \cos{\left(1 \right)} + 3 \]The solution.✓ Proved
Answer \( f(x) = - \frac{x^{4}}{3} - \frac{14 x}{3} + 3 x \sin{\left(1 \right)} + 3 \cos{\left(x \right)} - 3 \sin{\left(1 \right)} - 3 \cos{\left(1 \right)} + 3 \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 |
| 5 | ✓ 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 | the answer differentiated back to the given derivative, and the initial values checked |
Reviewers
gpt-oss:20b: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution misapplies the antiderivative: it uses \(-3\sin(1)\) instead of \(-3\cos(1)\) when evaluating \(f'(1)\), leading to an incorrect constant \(Cqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution misapplies the antiderivative: it uses \(-3\sin(1)\) instead of \(-3\cos(1)\) when evaluating \(f'(1)\), leading to an incorrect constant \(Cqwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04
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/antiderivative_ivp, checked 2026-10-04 with SymPy 1.14.0.