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Thread: Finding integral upper and lower bounds

  1. #1
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    Finding integral upper and lower bounds

    Hey guys,

    Here's an exercise I was looking at today.

    Use the Upper Bound Theorem to find an integral upper bound and the Lower Bound Theorem to find an integral lower bound of the zeros of $\displaystyle f(x)=x^3+3x^2-5x-10$

    Here was my explanation:

    According to the Upper Bound Theorem, c is an upper bound of the zeros of f(x) if
    when you divide f(x) by (x - c), there is no sign change in the quotient or remainder.

    So I picked a few possible factors of f(x). First I used (x - 1).

    The depressed polynomial produced was $\displaystyle x^2+4x-1-\frac{11}{x-1}$

    There is a sign change so 1 is not an upper bound.

    Next, I tried (x-2). The depressed polynomial was $\displaystyle x^2+5x+5$

    There was no sign change in the quotient or remainder so 2 is an upper bound.
    I contend that 2 is the greatest upper bound and there are no zeros of f(x) greater than 2.

    Now on to the lower bound.

    According to the Lower Bound Theorem, if c is an upper bound of the zeros of f(-x),
    then -c is a lower bound of the zeros of f(x).

    $\displaystyle f(-x)=-x^3+3x^2+5x-10$

    I went through the same process to find the upper bound for f(-x) as I did for f(x).

    Turns out that (x - 2) doesn't yield a quotient and remainder with no sign changes.

    Neither does (x - 3) or (x - 4). However, (x - 5) does.

    This means that all real zeros of f(x) can be found in the interval $\displaystyle -5 \leq x \leq 2$

    I agree with this, but is -5 the greatest lower bound?

    If you solve the thing algebraically, the roots are $\displaystyle \{\frac{-5 \pm \sqrt{5}}{2}, 2}\}$

    This shows the greatest (integral) lower bound is -4.

    Shouldn't all the zeros of f(x) be found on this interval: $\displaystyle -4 \leq x \leq 2 \:\:\text{?}$

    Since the exercise did not specify the words "greatest" or "least",
    maybe they were just looking for any upper and lower bound.
    I don't see the significance of that, though.
    If that were the case, the interval could be betwen -infinity and +infinity.
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  2. #2
    A Plied Mathematician
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    Well, I can't say I'm overly familiar with the Upper and Lower Bound Theorems, so I don't know if your reasoning is correct; however, you can see here that your bounds of -4 to 2 are correct.
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