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.
<span>The dimension of the rectangle would be length=#4/sqrt3# and width =4-#4/3= 8/3#
Explanation:
Let one vertex on the x axis be x, then the other vertex would be -x. Length of the rectangle would be thus 2x and width would be #4-x^2#
Area of the rectangle would be A=#2x(4-x^2)= 8x-2x^3#
For maximum area #(dA)/dx=0=8-6x^2#, which gives x=#2/sqrt3#
The dimension of the rectangle would be length=#4/sqrt3# and width =4-#4/3= 8/3#
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