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
f(x) = ax^3+bx^2+cx+d .
If a function is increasing, then its first derivative should be positive.
So f'(x) > 0.
f'(x) = (ax^3+bx^2+cx+d)' > 0.
f'(x) = 3ax^2+2bx+c > 0.
Therefore if a is positive an the discriminant (2b)^2 - 4(3a)c < 0, or 4b^2-12ac < 0 , Or b^2 - 3ac< 0 then for all x, f'(x) > 0 and f(x) is an increasing function.
Therefore if a is positive and if b^2-3ac < 0, then for all x f(x) is an increasing function.
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.
when would it then be decreasing from (-infinity to +infinity)?