Answer
Plot $IV$
Work Step by Step
The gradient vector filed is given as:
$∇f(x,y)=f_{x}(x,y)i+f_{y}(x,y)j$
$f(x,y)=x(x+y)=x^{2}+xy$
Here, $f_{x}(x,y)=2x+y$ and $f_{y}(x,y)=x$
Therefore, $∇f(x,y)=(2x+y)i+xj$
Points on the y-axis are the type of $(0,y)$.The vector filed for the points on the $y-axis$ is $F(0,y)=yi$ and the vectors are horizontal for the points on $y-axis$.
The vectors will point in negative x-direction when $y\lt 0$ and the vectors will point in positive y-direction when $y\gt 0$.
Hence, only plot $IV$ satisfies these conditions.
