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Proving that a nth degree polynomial is > (or < etc.) some constant.

  1. Jun 6, 2013 #1
    1. The problem statement, all variables and given/known data
    Let [itex]x\in[/itex]R,
    Prove that if x>2 then [itex]x^4 - 8x^3+24x^2-32x+16[/itex]

    2. Relevant equations


    3. The attempt at a solution
    So far I have only learned proofs involving even and odd numbers, that sort of thing. I'm not really sure how to approach this one. I was thinking that a proof by cases would suffice, so:

    Proof:
    Case I: Assume that x>2, and let S = (0,[itex]\infty[/itex]). If follows that,
    x = (m+2) for some m[itex]\in[/itex]S.

    And so,
    [itex]x^4 - 8x^3+24x^2-32x+16 = (m+2)^4 - 8(m+2)^3+24(m+2)^2-32(m+2)+16[/itex]

    but now I see the problem becoming too difficult. We haven't learned about epsilon-delta et al. yet. I can see that the conclusion is ALWAYS true, so really the proof is trivial, but I am not sure how to say that.
     
  2. jcsd
  3. Jun 6, 2013 #2

    Curious3141

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    Homework Helper

    Your question looks incomplete. Is that supposed to be [itex]x^4 - 8x^3+24x^2-32x+16 > 0[/itex]?

    What's ##(x-2)^4##? How does that help in your question?
     
  4. Jun 6, 2013 #3
    Thank you for the reply,

    Sorry, yes the whole equation should be greater than zero, or equal to/less than something, I just made the question up as an example.

    I see that (x-2)^4 is in fact equal to that polynomial. Therefor, for it is always going to be greater than zero because it is to an even power (and there is the case where x is 2, and then it IS equal to zero). I don't know how I am supposed to write this though? I'm aware that the conclusion is true for all x, but how would I even prove that:

    [itex](x-2)^4 > 0[/itex], for some x[itex]\in[/itex]R

    Edit:

    Oh wait, I think I see what you are getting at...
     
    Last edited: Jun 6, 2013
  5. Jun 7, 2013 #4

    Curious3141

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    That should be ##(x-2)^4 \geq 0## for some ##x \in \mathbb{R}##. Equality occurs at x = 2. These little details are important.

    You don't even need to bother with the case ##x \leq 2## since the question didn't ask for it.

    Sketch the curve, and just make the appropriate arguments.

    To prove the function is always increasing beyond x = 2, what can you say about the derivative for any x greater than 2?
     
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