Arectangle has one side on the x-axis and two vertices on the curve, = 1/1+x^2. find the coordinates of the vertices of the rectangle with the maximum area

I believe that you have issues with the function for the curve. Because if that function is correct, the rectangle with the maximum area will have the two coordinates off the X-axis of (-infinity, infinity) and (infinity, infinity) and the area of the rectangle will be infinite.

Assuming correct expression is
y = 1/1 + x^2 = 1 + x^2.
Then you need to find 2 x values that have the same y value. You'll quickly realize that the values X and -X will work and give you the same Y value. And as you use large absolute values of X, the Y value will also increase. And carried to the logical limit, the largest possible rectangle will happen with X values of -infinity and +infinity.

Assuming correct expression is
1 = 1 + x^2, which simplifies to 0 = x^2, which has the exact same argument. The coordinates of the 2 points are (-infinity, infinity) and (infinity, infinity). So once again, the area of the rectangle increases without limit.