Question

If P$(x_1,y_1)$, Q$(x_2,y_2)$, R$(x_3,y_3)$ be the points of inflection of the curve $x^{2}y-x+y-1=0,$ then number of rational vertices of the polygon PQR is

Answer 1

$y=\frac{x+1}{x^2+1}$
$\frac{dy}{dx}=\frac{(-x^2-2x+1)}{(x^2+1{)}^2}$
$\frac{d^{2}y}{d{x}^2}=\frac{2(x-1)(x^2+4x+1)}{(x^2+1{)}^3}\Rightarrow x=1,x=-2+\sqrt{3},-2-\sqrt{3}$
For these three points $f^{\prime \prime }\left(x\right)\text{vanishes and}{f}^{\prime }\left(x\right)\text{is defined}$ so only one rational point.