Consider the function $f(x)=mx^2+(2m-1)x+(m-2)$. Choose the correct options:
$(A).$ If $f(x)>0$ for all $x \in R$, then $m \in (- \infty, -1/4)$
$(B).$ The number of integral values of $m$ greater than $-5$ so that $f(x)<0$ for all $x \in R$ are 4.
$(C).$ The number of integral values of $m$ less than $50$ so that roots of the quadratic equation $f(x)=0$ are rational are 6.
$(D)$ The curve $y=f(x)$ touches the $x-axis$ if $m=-1/4$
I have found that option($2,4$) are correct and $1$ is wrong. I need help with option $3$. I know for rational roots Discriminant should be square of a rational number and coefficients should be rational too but not able to apply it here.