Question

The least positive integral value of $a$ for which the equation $x^2-2(a-1)x+2a+1=0$ has both roots positive is

Answer 1

Since, both roots are positive, required conditions are
$\left(i\right)D\ge 0,\left(ii\right)-\frac{b}{2a}\ge 0,\left(iii\right)f\left(0\right)>0$

$\left(i\right)D\ge 0\Rightarrow (a-1{)}^2-(2a+1)\ge 0$
$\Rightarrow a\in (-\infty ,0]\cup [4,\infty )$

$\left(ii\right)\frac{-b}{2a}>0\Rightarrow a>1$

$\left(iii\right)f\left(0\right)>0\Rightarrow a>-\frac{1}{2}$

$\left(i\right)\cap \left(ii\right)\cap \left(iii\right)\Rightarrow a\ge 4$
So,least positive integer is $4.$