At the right is the slope field for yfxy Based on the graph

At the right is the slope field for y?=f(x,y). Based on the graph, identify where the function f(x,y) is positive, negative, and zero. Is f(x,y) a function of x alone, y alone, or a function of both variables together? Find a function f(x,y) whose slope field looks like this.

Solution

Clearly f(x,y) is just a function of x . Its independent of y .

For a given x f(x,y) is constant in y direction.

f(x,y) = 0 at x=3,-3

f(x,y) < 0 when |x| < 3

f(x,y) > 0 when |x| > 3

f(1,1) = -8

f(x,y ) = (|x|-3)^3/|x| is one such function

zero