Maximum rate of change
Problem 10.550 · medium
Find the maximum rate of change of \( \displaystyle f(x, y) = \sin{\left(x y \right)} \) at \( \displaystyle (-1, 2) \), and the direction in which it occurs.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sin{\left(x y \right)}\\\frac{\partial}{\partial y} \sin{\left(x y \right)}\end{matrix}\right] = \left[\begin{matrix}y \cos{\left(x y \right)}\\x \cos{\left(x y \right)}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}2 \cos{\left(2 \right)}\\- \cos{\left(2 \right)}\end{matrix}\right] \]∇f(-1, 2).✓ Proved
- \[ - \sqrt{5} \cos{\left(2 \right)} \]The maximum rate of change is ‖∇f‖, in the direction of ∇f.✓ Proved
Answer \( \|\nabla f\| = - \sqrt{5} \cos{\left(2 \right)}\ \text{in the direction of}\ \left\langle 2 \cos{\left(2 \right)}, - \cos{\left(2 \right)} \right\rangle \)
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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the largest directional difference quotient over 3600 directions |
Reviewers
gpt-oss:20b: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The maximum rate of change is the magnitude of the gradient, which is \u221a5\,|\cos2|, not \(-\u221a5\cos2\). The sign of the norm cannot be negative; thqwen3.6:27b-mlx: fail (error) — The solution incorrectly equates the maximum rate of change with the signed magnitude -sqrt(5)*cos(2). The maximum rate of change is the norm of the gradient, which must be non-negative. Since cos(2) is negative, the norm is sqrt(5)*cos(2) (positive), but the solution presents a negative value. Additionally, the direction vector provided is not a unit vector, which is typically required when stating the direction of maximum increase.
Every verdict on record (4)
gpt-oss:20b: inconclusive 2026-10-13 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The maximum rate of change is the magnitude of the gradient, which is \u221a5\,|\cos2|, not \(-\u221a5\cos2\). The sign of the norm cannot be negative; thqwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution incorrectly equates the maximum rate of change with the signed magnitude -sqrt(5)*cos(2). The maximum rate of change is the norm of the gradient, which must be non-negative. Since cos(2) is negative, the norm is sqrt(5)*cos(2) (positive), but the solution presents a negative value. Additionally, the direction vector provided is not a unit vector, which is typically required when stating the direction of maximum increase.gpt-oss:20b: inconclusive 2026-10-13 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly states the maximum rate of change as \(-\sqrt5\cos2\). Since \(\cos2<0\), this expression is positive, but the magnitude of the gqwen3.6:27b-mlx: fail (style) 2026-10-13 — [domain objection, downgraded to style] The maximum rate of change is defined as the magnitude of the gradient, which must be non-negative. The solution incorrectly equates the maximum rate of change to a negative value (-sqrt(5)*cos(2)), failing to take the absolute value of the gradient's magnitude.
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/max_rate_of_change, checked 2026-10-13 with SymPy 1.14.0.