# 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

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### 3 Answers

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)?