Assuming $\beta > \alpha $ which of the following option(s) is/are correct?

(A) $y \le \alpha $
(B) $y \ge \beta $
(C) $\alpha \le y \le \beta $
(D) $y \in \mathbb{R}$

Solution

${x^2} – 2xy + [(\alpha + \beta )y – \alpha \beta ] = 0$

$\because x \in \mathbb{R},\Delta \ge 0$

$\therefore 4{y^2} – 4[(\alpha + \beta )y – \alpha \beta ] \ge 0$

$ \Rightarrow {y^2} – \alpha y – \beta y + \alpha \beta \ge 0$

$ \Rightarrow (y – \alpha )(y – \beta ) \ge 0$

The wave-curve is shown below for the expression $ (y – \alpha )(y – \beta ) $ assuming $\beta > \alpha $.

Hence, (A) & (B).