Thread: concepts (need experts to confirm)

1. concepts (need experts to confirm)

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.

2. What about $\displaystyle x^3+a$ ?
Or even $\displaystyle x^3+x^2+x+1$ ?

No pattern.

3. Originally Posted by stupidguy
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.
For a cubic $\displaystyle f(x)=ax^3+bx^2+cx+d$ where $\displaystyle a\ne0$, we can make some observations (provable)

1) there is at least one real root
2) there are at most three real roots
3) if a > 0, then $\displaystyle \displaystyle \lim_{x\to\infty} f(x)=\infty$ and $\displaystyle \displaystyle \lim_{x\to-\infty} f(x)=-\infty$
4) if a < 0, then $\displaystyle \displaystyle \lim_{x\to\infty} f(x)=-\infty$ and $\displaystyle \displaystyle \lim_{x\to-\infty} f(x)=\infty$

You may not understand (3) and (4).... anyway for the example you gave

x(x+1)(x-1)<0

if you look at the equality (instead of inequality)

x(x+1)(x-1)=0

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

$\displaystyle (-\infty, -1)$, $\displaystyle (-1, 0)$, $\displaystyle (0, 1)$, $\displaystyle (1, \infty)$

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 $\displaystyle (-\infty, -1)$ and $\displaystyle (0, 1)$.

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)

4. Originally Posted by Moo
What about $\displaystyle x^3+a$ ?
Or even $\displaystyle x^3+x^2+x+1$ ?

No pattern.
Quite true. It only applies to y=(x-a)(x-b)(x-c), right?

5. Originally Posted by stupidguy
Quite true. It only applies to y=(x-a)(x-b)(x-c), right?
With a,b,c being real, yes.

So just use the usual method If there was a simpler one, I'm sure all high school teachers would've let all of their students know !

6. Originally Posted by undefined
.......
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)
Amazing. Undefined has predicted all the qn i am about to ask.

7. Is it true that all cubic functions look like a squiggle. "~" .

8. Originally Posted by stupidguy
Is it true that all cubic functions look like a squiggle. "~" .
That depends very much on how strictly we define "looks like."

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)

9. As another example of a non-squiggly cubic, you might consider

$\displaystyle x^3 + 3x$

10. Instead of "squiggle", I prefer to refer to cubics as "Loch Ness monsters"....