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  1. #1
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    Question concepts (need experts to confirm)

    I would like to verify if cubic functions always have the following pattern

    concepts (need experts to confirm)-untitled.jpg.

    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.
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  2. #2
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    What about x^3+a ?
    Or even x^3+x^2+x+1 ?

    No pattern.
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  3. #3
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    Quote Originally Posted by stupidguy View Post
    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 f(x)=ax^3+bx^2+cx+d where 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 \lim_{x\to\infty} f(x)=\infty and \displaystyle \lim_{x\to-\infty} f(x)=-\infty
    4) if a < 0, then \displaystyle \lim_{x\to\infty} f(x)=-\infty and \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

    (-\infty, -1), (-1, 0), (0, 1), (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 (-\infty, -1) and (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)
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  4. #4
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    Quote Originally Posted by Moo View Post
    What about x^3+a ?
    Or even x^3+x^2+x+1 ?

    No pattern.
    Quite true. It only applies to y=(x-a)(x-b)(x-c), right?
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  5. #5
    Moo
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    Quote Originally Posted by stupidguy View Post
    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 !
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  6. #6
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    Quote Originally Posted by undefined View Post
    .......
    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.
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  7. #7
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    Is it true that all cubic functions look like a squiggle. "~" .
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  8. #8
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by stupidguy View Post
    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)
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  9. #9
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    As another example of a non-squiggly cubic, you might consider

    x^3 + 3x
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  10. #10
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    Instead of "squiggle", I prefer to refer to cubics as "Loch Ness monsters"....
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