Answer
$\dfrac{\sqrt {17}}{2}, \lt 0,\dfrac{1}{2},2 \gt$
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: $f(x,y,z)=x \ln (yz)$
$\nabla f(x,y,z)=\lt \ln (yz)y , x/y,x/z \gt $
From the given data, we have $f(x,y,z)=f(1,2,\dfrac{1}{2})$
$\nabla f(1,2,\dfrac{1}{2})=\lt \ln 1, 1/2,2 \gt=\lt 0,\dfrac{1}{2},2 \gt$ $|\nabla f(1,2,\dfrac{1}{2})|=\sqrt{0^2+(\dfrac{1}{2})^2+2^2}=\dfrac{\sqrt {17}}{2}$
Therefore, the maximum rate of change of $f(x,y)$ and the direction is: $\dfrac{\sqrt {17}}{2}, \lt 0,\dfrac{1}{2},2 \gt$
