Discussion
For \(f = x^{4} - 5xy^{2}\), the direction of maximum increase of \(f(x,y)\) at the point (2, 2) is along
Answer: C
The direction is along \(3\hat{i} - 10\hat{j}\).
A function increases fastest along its gradient:
\[ \begin{aligned} \nabla f &= (4x^{3} - 5y^{2})\hat{i} - 10xy\,\hat{j} \\ \nabla f(2,2) &= (32 - 20)\hat{i} - 40\hat{j} \\ &= 12\hat{i} - 40\hat{j} = 4(3\hat{i} - 10\hat{j}) \end{aligned} \]
A direction is fixed only up to a positive multiple, so \(3\hat{i} - 10\hat{j}\) is the same direction. \(-12\hat{i} + 40\hat{j}\) points the opposite way, the direction of fastest decrease, and the other two options have a wrong sign on one component.
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