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Thread: Finding Real Roots

  1. #1
    Junior Member
    Oct 2009
    Bloomington, IN

    Finding Real Roots

    Hello, and thanks for the help in advance!

    I have several problems that are all in the same vein as the one below, but I don't have much of an idea as to how to solve them. I would apply the quadratic formula if exponents allowed, because that is how I am accustomed to finding roots, but these problems don't seem to allow for that.

    Show that the following equation has exactly one real root:

    1 + 2x + x^3 + 4x^5 = 0

    Thanks again for the only other idea is to take the derivative of both sides, but I'm not sure where this would lead us...
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  2. #2
    Junior Member
    Oct 2009
    Hint: Letting $\displaystyle f(x)=4x^5+x^3+2x+1,$ show that $\displaystyle f'(x)>0$ for all $\displaystyle x\in\mathbb R.$ What does this tell you about $\displaystyle f(x)\,?$
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