What about ?
Or even ?
I would like to verify if cubic functions always have the following pattern
And also, does it always have a positive-negative-positive-negative or negative-positive-negative-positive pattern.
These facts are essential to help me solve inequalities with cubic polynomials. Eg. x(x+1)(x-1)<0.
As long as I establish the sign before 1 root. I can deduce the pattern, hence saving much of my time.
I need at least 3 experts to verify.
1) there is at least one real root
2) there are at most three real roots
3) if a > 0, then and
4) if a < 0, then and
You may not understand (3) and (4).... anyway for the example you gave
if you look at the equality (instead of inequality)
you should be able to see immediately there are three real roots, 0, -1 and 1. Thus we pay attention to four regions of the real number line, which are
, , ,
Now you can test out values in each interval, or you can use (3) above and a little logic to deduce that the satisfying intervals are and .
Note that if a cubic has three real roots then the function must change signs on either side of each root (locally); if it has two real roots then one must be a double root (the function does not change sign there); if one real root then the function changes sign there.
Example of two real roots: f(x) = (x-2)(x-3)^2 (graph it to get a visual)
Example of one real root: f(x) = (x-5)^3 (graph it) -- this one has a triple root
Another example of one real root: f(x) = (x-2)(x-3)(x-4) + 20 (this does not have a triple root)
Some possible counter-examples:
f(x)=x^3 (it only changes direction once)
f(x)=-(x-1)(x-2)(x-3) (it is upside down)
f(x)=10x(x-1)(x+1) (if you use the same scale for x- and y-axes, it will be a very "sharp" squiggle)