Calculus - Increasing function Problem Find all values for $a\in \mathbb R$ so the function  $\, f(x) = x^3 + ax^2 + 3x - 1\,$ is always increasing in $\mathbb R$:
$\ f'(x) = 3x^2 + 2ax + 3 $ , So for the function to be increasing, $\,f'(x) $ must be greater than $ 0.$
Therefore $ a\gt -3(x^2+1)/2x $
Is this right for an answer ? Thank you.

You should think about whether dividing by x was actually allowed. Remember that x could be 0, since you need that inequality to hold for all real x. For $3x^2+2ax+3$ to be above the x axis, it cannot have any real roots. So it must have this quantity $b^2-4ac$ (the discriminant) be negative. You know what b and c are and you can solve for a. 
