Answer
$\sqrt 5,\lt 2,0,1 \gt$
Work Step by Step
Our aim is to determine the maximum rate of change of $f(x,y)$. In order to find this, we have : $D_uf=|\nabla f(x,y,z)|$
Formula to calculate the directional derivative: $D_uf=\nabla f(x,y,z) \cdot u$
Given: $f(x,y,z)=ze^{xy}$
$\nabla f(x,y,z)=\lt zye^{xy},zxe^{xy},e^{xy} \gt$
From the given data, we have $f(x,y,z)=f(0,1,2)$
$\nabla f(0,1,2)=\lt zye^{xy},zxe^{xy},e^{xy} \gt=\lt 2,0,1 \gt$
$|\nabla f(0,1,2)|=\sqrt{2^2+0^2+(1)^2}=\sqrt 5$
Therefore, the maximum rate of change of $f(x,y)$ and the direction is: $\sqrt 5,\lt 2,0,1 \gt$
