Answer
$\sqrt{17}, \lt 4,1 \gt$
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: $f(s,t)=te^{st}$
$\nabla f(x,y)=\lt t^2 e^{st},e^{st}+ste^{st} \gt $
From the given data, we have $f(x,y)=f(0,2)$
Thus, $\nabla f(0,2)=\lt 2^2 e^{0},e^{0}+s(0) \gt=\lt 4,1 \gt$
or, $|\nabla f(0,2)|=\sqrt{4^2+ 1^2}=\sqrt{17}$
Therefore, the maximum rate of change of $f(x,y)$ and the direction is:$\sqrt{17}, \lt 4,1 \gt$
