Answer
$\sqrt 6, \lt -1,-1,-2 \gt$
Work Step by Step
Formula to calculate the maximum rate of change of $f$: $D_uf=|\nabla f(x,y)|$
$\nabla f(x,y)=\lt \dfrac{1}{z}, \dfrac{1}{z}, -\dfrac{x+y}{z^2}\gt $
$\nabla f(1,1,-1)=\lt \dfrac{1}{-1}, \dfrac{1}{-1}, -\dfrac{1+1}{(-1)^2}\gt=\lt -1,-1,-2 \gt$
$|\nabla f(0,1)|=\sqrt{(-1)^2+(- 1)^2+(-2)^2}=\sqrt 6$
Hence, the required answers are:
$\sqrt 6, \lt -1,-1,-2 \gt$
