#### Answer

All the points on the line $y=x+1 $

#### 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: $f(x,y)=x^2+y^2-2x-4y$
$\nabla f(x,y)=\lt 2x-2,2y-4 \gt$
We have found the direction vector as $\lt 1,1 \gt$
or, $\lt 2x-2,2y-4 \gt=n\lt 1,1 \gt$
Hence, $n=2x-2, n=2y-4$ and $2x-2=2y-4$
Therefore, the points at which the direction of fastest change are:All the points on the line $y=x+1 $