Answer
$\dfrac{3}{5},\lt \dfrac{2}{5},\dfrac{1}{5},\dfrac{2}{5} \gt$
Work Step by Step
Formula to calculate the maximum rate of change of $f$: $D_uf=|\nabla f(x,y)|$
$\nabla f(x,y)=\lt \dfrac{qr}{1+(pqr)^2},\dfrac{pr}{1+(pqr)^2},\dfrac{pq}{1+(pqr)^2} \gt$
$\nabla f(1,2,1)=\lt \dfrac{2}{5},\dfrac{1}{5},\dfrac{2}{5} \gt$
$|\nabla f(1,2,1)|=\sqrt{(\dfrac{2}{5})^2+(\dfrac{1}{5})^2+(\dfrac{2}{5})^2}=\sqrt {\dfrac{9}{25}}=\dfrac{3}{5}$
Hence, the required answers are:
$\dfrac{3}{5},\lt \dfrac{2}{5},\dfrac{1}{5},\dfrac{2}{5} \gt$
