Answer
$$\sqrt{65}, \lt 1, 8 \gt$$
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 : $D_uf=|\nabla f(x,y)|$
Given: $f(x,y)=4y \sqrt x$
$\nabla f(x,y)=(\dfrac{2y}{\sqrt x},4\sqrt x)$
From the given data, we have $f(x,y)=f(4,1)$
Thus, $\nabla f(4,1)=\lt \dfrac{2(1)}{\sqrt (4)},4\sqrt (4) \gt=\lt 1, 8 \gt$
or, $|\nabla f(4,1)|=\sqrt{1^2+ 8^2}=\sqrt{65}$
Therefore, the maximum rate of change of $f(x,y)$ and the direction is: $\sqrt{65}, \lt 1, 8 \gt$
