increasing functions using discriminant of first derivitive.

The derivative of the cubic represents its gradient, and the derivative is a quadratic.

If I assume the cubic is a positive cubic, then it's derivative is a positive quadratic. So the derivative will be "U" shaped. Therefore the gradient come down from infinity, comes down to a minimum, then goes back up to infinity.

The discriminant is used to tell us exactly how far down this positive quadratic goes. A positive discriminant means that there must exist roots to the quadratic (I.e. there are points where the quadratic is equal to zero), whereas a negative discriminant means that there is no point on the quadratic that is equal to zero (I.e. The quadratic comes down to some minimum value greater than zero, then goes back up again).

So if the discriminant of a positive cubic is negative, then the cubic is increasing for all x (as its derivative must be above 0 for all x). Whereas if the discriminant is positive, then the cubic is increasing, then decreasing, then increasing again. The points where is changes from increasing to decreasing or vice versa will be when the quadratic is equal to zero.

Similar logic applies to a negative cubic but with opposite outcomes.