Answer
$\dfrac{3}{5},\lt \dfrac{2}{5},\dfrac{1}{5},\dfrac{2}{5} \gt$
Work Step by Step
Our aim is to determine the maximum rate of change of $f(x,y,z)$.In order to find this, we have : $D_uf=|\nabla f(x,y,z)|$
Given: $f(p,q,r)=arctan(pqr)$
$\nabla f(x,y)=\lt \dfrac{qr}{1+(pqr)^2},\dfrac{pr}{1+(pqr)^2},\dfrac{pq}{1+(pqr)^2} \gt$
From the given data, we have $f(p,q,r)=f(1,2,1)$
Thus, $\nabla f(1,2,1)=\lt \dfrac{2}{5},\dfrac{1}{5},\dfrac{2}{5} \gt$
or, $|\nabla f(1,2,1)|=\sqrt{(\dfrac{2}{5})^2+(\dfrac{1}{5})^2+(\dfrac{2}{5})^2}=\sqrt {\dfrac{9}{25}}=\dfrac{3}{5}$
Therefore, the maximum rate of change of $f(x,y)$ and the direction is:$\dfrac{3}{5},\lt \dfrac{2}{5},\dfrac{1}{5},\dfrac{2}{5} \gt$
