( cont. of my post #2 )
Let us name
Graph 1 : ##x^5-x^2+y^3-y^2=0##
Graph 2 : ##x(x+y)=2##
The x coordinate of the crossing point satisfies
(x-1)(x^7+x^6-x^4+6x^3+6x^2-8x-8)=0
So we now (x,y)=(1,1) is a crossing point, let us say it P.
At P by simple calculation
the gradient of graph 1 = the gradient of graph 2 = -3
The second derivative of graph 1 = 4
The second derivative of graph 2 = -54
So we show the graph 2 surpass graph 1 around point P.
This is just a local discussion around P.
The graph of the remaining part of the above crossing point equation
So as for x < 1.083... where the red dot point in the graph, we shall not have another crossing which goes with the statement.
The red point should not correspond to another crossing but the x bounded point, but I would like to get a confirmation about my interpretation.
Anyway, the above three lines I said watching the graph are not proof but a speculation.
For graph 1
\frac{dy}{dx}=-\frac{x(5x^3-2)}{y(3y-2)}
dy/dx=0 at x=0, ##(\frac{2}{5})^{1/3}##, i.e. (0,1), ##((\frac{2}{5})^{1/3},\beta)##
where ##\beta^3+\beta^2+(2/5)^{5/3}+(2/5)^{2/3},\beta>0##
dx/dy=0 at y=0, 2/3, i.e. ##(1,0),(\alpha,2/3)## where
##\alpha^5-\alpha^2-4/27=0,\alpha>0##
For graph 2
\frac{dy}{dx}=\frac{\frac{x}{\sqrt{x^2+4}}-1}{2} <0
So y is monotonously decreasing function of x
for x=0, y=##+\infty## >1
for x=##(\frac{2}{5})^{1/3}, y=\frac{\sqrt{(2/5)^{2/3}+4}-(2/5)^{1/3}}{2}## > ##\beta## by numerical calculation
for y=0, x=##\sqrt{2}## > 1
for y=2/3, x=##\frac{2\sqrt{19}-1}{3} > \alpha## by numerical calculation
That means graph 2 surpass graph 1 in region x,y>0.
Thus I wish the statement was proved, though I would like to know a more elegant solution very much.