Method to determine type of stationary point without calculating second derivative?

Question

I have the function $$y= \frac{c+dx^2}{2(bx-a)}$$ where a,b,c,d are real constants and $c,d > 0$ . I have calculated it’s stationary points to be $$x=\frac{a \pm \sqrt{a^2+b^2 \frac{c}{d}}}{b}$$ . I want to determine which one is the maximum/minimum without calculating the second derivative or plotting a graph when fixing the constants a,b,c,d. Is there any way I can do this?

Answer 1

Yes, you can. If $b>0$ then for $x\to \infty $ the $y$ increases, thus at $x={a+...\over b}$ you have local minimum and at the second stationary point you have local maximum.

If $b<0$ you have to reverse all I said before. And if $b=0$ you have a quadratic equation function...