Maximum rate of change
Problem 10.329 · medium
Find the maximum rate of change of \( \displaystyle f(x, y) = x^{2} + x y + y^{2} \) at \( \displaystyle (-1, 2) \), and the direction in which it occurs.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + x y + y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} + x y + y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x + y\\x + 2 y\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}0\\3\end{matrix}\right] \]∇f(-1, 2).✓ Proved
- \[ 3 \]The maximum rate of change is ‖∇f‖, in the direction of ∇f.✓ Proved
Answer \( \|\nabla f\| = 3\ \text{in the direction of}\ \left\langle 0, 3 \right\rangle \)
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: fail (misleading) — The solution states that the direction of maximum increase is the vector ⟨0,3⟩. The direction should be given as a unit vector (⟨0,1⟩), since the gradient’s direction is determined up to scaling. This could mislead a student into thinking any scalar multiple of the gradient is an acceptable direction.qwen3.6:27b-mlx: pass — The solution correctly identifies the gradient, evaluates it at the specified point, and applies the theorem that the maximum rate of change is the magnitude of the gradient in the direction of the gradient. The algebraic steps are verified as correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the gradient, evaluates it at the specified point, and applies the theorem that the maximum rate of change is the magnitude of the gradient in the direction of the gradient. The algebraic steps are verified as correct.gpt-oss:20b: fail (misleading) 2026-10-04 — The solution states that the direction of maximum increase is the vector ⟨0,3⟩. The direction should be given as a unit vector (⟨0,1⟩), since the gradient’s direction is determined up to scaling. This could mislead a student into thinking any scalar multiple of the gradient is an acceptable direction.qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: fail (misleading) 2026-10-04 — The solution states the direction of maximum increase as the vector ⟨0,3⟩, which is not a unit vector. The correct direction is the unit vector (0,1).
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 2026-10-04 with SymPy 1.14.0.