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A function

  1. Jun 21, 2008 #1
    1. The problem statement, all variables and given/known data

    [tex]f(x)=x^{12}-x^{9}+x^{4}-x+1[/tex]

    Find th minimum possible value of the function, and show that its an always increasing function.
    Actually I want to prove that f(x)>0 for all x.

    3. The attempt at a solution
    Its difficult for me to factorize this.
    [tex]dy/dx=12 x^{11} - 9 x^{8} + 4 x^{3} -1 [/tex]
    How shall I find the required information from the derivative. Is there any other way?
     
  2. jcsd
  3. Jun 21, 2008 #2

    kreil

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    Graphing the fcn with a calculator is the easiest way to solve this.

    - f'(x) = 0 at x = 0.67460257.....
    - f'(x) monotonically increases, but is not always positive.
    - f'(x) changes from - to + through the zero listed above (so f(x) achieves an absolute min at x = 0.6746....)
    - This minimum is y = 0.5124....

    It follows from the statements above that since f(x) has 1 extremum, a minimum, and since this minimum is positive that f(x) > 0 for all x.
     
  4. Jun 21, 2008 #3
    I did this a little differently. We start by plugging two values of x, a and -a, where a>0. From this we have

    [tex]f(-a)=a^{12}+a^9+a^4+a+1>0[/tex] and
    [tex]f(a)=a^{12}-a^9+a^4-a+1[/tex]​

    Now, since [tex]a^{12}>a^9[/tex] and [tex]a^4>a[/tex], [tex]f(a)>0[/tex]. Since f(-a) and f(0) are both obviously always going to be positive, we can conclude that [tex]\forall x \in \mathbb{R}, f(x)>0[/tex]. Q.E.D.
     
  5. Jun 22, 2008 #4
    You are right exactly! But the problem is friend that we are not allowed to use the calculator. :smile:
     
  6. Jun 22, 2008 #5
    Thanks a lot foxjwill. But what u said is not true for 0<x<1
    [tex]a^{12}>a^9[/tex] and [tex]a^4>a[/tex]
    These inequalities are not true for the interval x[tex]\in[/tex](0,1)
     
  7. Jun 22, 2008 #6

    matt grime

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    Could you clarify what you mean by 'an always increasing function', since with my reasonable assumption of what that means, that polynomial is not an "always increasing function" on the real line, since the derivative at 0 is negative.
     
  8. Jun 22, 2008 #7

    Gib Z

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    First of all, let us factor the function is this manner - [tex]f(x) = x^{12} - x^9 + x^4 - x +1 = x^9( x^3 -1) + x( x^3-1) + 1 = (x^9 + x) (x^3 -1) + 1 = x(x^8 + 1) (x-1)(x^2 +x +1) +1 [/tex].

    Up to that point, it should all be obvious steps. Now, let us take those 2 linear terms to make one quadratic, and also take the other quadratic, and complete the square on both of them:

    [tex]f(x) =(x^2 - x)(x^2+x+1)(x^8 +1) + 1 = \left( \left(x- \frac{1}{2}\right)^2 -\frac{1}{4} \right) \left( \left(x+\frac{1}{2} \right)^2 +\frac{3}{4}\right) (x^8 +1) + 1 [/tex]

    From there, expand the first two brackets as such:

    [tex]f(x) = \left[ ( x-\frac{1}{2})^2 \left( (x+\frac{1}{2})^2 + \frac{3}{4} \right) - \frac{1}{4} \left( (x+\frac{1}{2})^2 + \frac{3}{4} \right) \right] (x^8+1) + 1[/tex]
    To get the minimum value of the function, we first must find the minimum value of the terms in the square brackets; this means we want the first term to be as small as possible, and our second completed square to be as large as possible. For the positive term, the minimum value of the perfect square in [0,1] is 1/4, whilst the minimum value of the completed square is 1. Hence, their product can never be less than 1/4.

    For the negative term, we maximize the completed square in [0,1] to to see our negative term has at most a magnitude of 3/4. Hence, the minimum value of the square brackets in -1/2. Since we want to subtract as much as possible, maximize the degree 8 term in (0,1). That term will always be less than 2. Hence we will always subtract less than 1 from the remaining constant term if x (0,1), and if we let x=0 and x=1 into f(x) we see that f(x) =1. And so f(x) will always be greater than zero in [0,1]


    EDIT: Not only did I express that horribly, I'm certain there has got to be a more elegant way than that.
     
    Last edited: Jun 22, 2008
  9. Jun 22, 2008 #8
    Reference in rotation among the transactions
    +,-,+,-,+

    if f(r) =0 then r>0
    r=root in f
     
  10. Jun 22, 2008 #9

    matt grime

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    Are you trying to say that the roots of, say, x^3-x must all be strictly positive, m_s_a? Or perhaps you should explain your cryptic comment?
     
  11. Jun 22, 2008 #10
    Yes .. Yes
    This is what I want to say
     
  12. Jun 22, 2008 #11

    matt grime

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    So, if x^3=x, then x must be strictly positive? Even though (-1)^3=-1 and 0^3=0? And indeed for any polynomial with zero constant term 0 is always a root irrespective of signs.
     
  13. Jun 22, 2008 #12
    Challenge
    write Polynomail
    More than double the rotation
    an=/=0
    r=0 or A negative number
     
    Last edited: Jun 22, 2008
  14. Jun 22, 2008 #13

    matt grime

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    I'm afraid I cannot understand what you've written. That's why I asked you to clarify what you mean. You really need to define 'rotation' and then '(more than) double rotation'. One can only guess that you're referring to the signs of the coefficients changing, and as I pointed out to you something like

    x^6 - x^5 + x^4 - x^3 + x^2 - x

    has a root at x=0, which seems to contradict what you *might* be saying.
     
  15. Jun 22, 2008 #14
    an=/=0
    Because
    Rotation does not appear in a double -
     
    Last edited: Jun 22, 2008
  16. Jun 22, 2008 #15
    Try
    Term is not defined
    Time is Open:devil:
     
  17. Jun 22, 2008 #16
    I think U r wrong there. The terms inside the square bracket:
    the minimum of first is 0
    the maximum of the term to be subtracted is 3/4
    and the max value of [tex]x^{8}+1[/tex] is two
    which on multiplication gives< -1.
    SO I am still having a problem with this.
     
  18. Jun 22, 2008 #17

    matt grime

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    Does anyone know what m_s_a is on about?

    Since you (m_s_a) can't be explain yourself I'm going to stop trying to understand.
     
  19. Jun 22, 2008 #18
    Hey guys! I am unable to understand a word u say. Please be more explicit, and dont deviate from the original question :)
    regards,
    Ritwik
     
  20. Jun 22, 2008 #19

    matt grime

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    I believe that m_s_a was attempting to shed some light onto something... I have no idea what though.
     
  21. Jun 23, 2008 #20

    Gib Z

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    Ahh God damn it.

    [tex]f(x) = (x^4 - x^9) + (-x + x^{12} ) + 1[/tex].
     
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