for the cubic: f(x) = ax^3+bx^2+cx+d, a is not zero, find conditions on a,b,c and d to ensure that: a) f is always increasing on (-infinity, +infinity

For f(x) to be increasing, the derivative of f(x) has to be
positive.

We'll determine the first
derivative:

f'(x) = 
(ax^3+bx^2+cx+d)'

f'(x) = 3ax^2 + 2bx +
c

For the expression of the first derivative to be positive, we'll
impose the constraint that the discriminant delta to be
negative.

delta = (2b)^2 - 4*3a*c

delta
= 4b^2 - 12ac

delta < 0

4b^2 -
12ac < 0

We'll divide by 4:

b^2
- 3ac < 0

We'll add 3ac both
sides:

b^2 <
3ac

The constraint for f(x) to be increasing over the
interval (-infinite, +infinite) is that: b^2 < 3ac.