Question

If the point $(\alpha ,{\alpha }^2)$ lies on or inside the triangle formed by the lines $x^{2}y+x{y}^2-2xy=0,$ then the largest possible value of $\alpha$ is

Answer 1

$x^{2}y+x{y}^2-2xy=0$
$\Rightarrow \left(xy\right)(x+y)-2xy=0$
$\Rightarrow xy(x+y-2)=0$

Point lies on or inside the triangle formed by the lines
$x=0,y=0\text{and}x+y=2$
Required conditions are
$\left(i\right)\alpha \ge 0\left(ii\right){\alpha }^2\ge 0$
$\left(iii\right)\alpha +{\alpha }^2-2\le 0$

$\Rightarrow \alpha \in [0,1]$

So, largest value of $\alpha =1$