Answer
$\sqrt{65}, \lt 1, 8 \gt$
Work Step by Step
Formula to calculate the maximum rate of change of $f$: $D_uf=|\nabla f(x,y)|$
$\nabla f(x,y)=(\dfrac{2y}{\sqrt x},4\sqrt x)$
$\nabla f(4,1)=\lt \dfrac{2(1)}{\sqrt (4)},4\sqrt (4) \gt=\lt 1, 8 \gt$
$|\nabla f(4,1)|=\sqrt{1^2+ 8^2}=\sqrt{65}$
Hence, the required answers are:
$\sqrt{65}, \lt 1, 8 \gt$
