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.
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