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Math Help - prove that this equation has 1 real root

  1. #1
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    prove that this equation has 1 real root

    y=x^3+8x-19

    I know the derivative is 3x^2+8, but how does this tell me there is no turning point and therefore that the function is increasing has one root

    thanks

    Bryn
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  2. #2
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    Make it of the form (x+a)(bx^2 + cx + d)

    Then prove bx^2 + cx + d has no real roots.
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  3. #3
    MHF Contributor red_dog's Avatar
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    f(x)=x^3+8x-19, \ f'(x)=3x^2+8>0, \ \forall x\in\mathbf{R}
    So, f is strictly increasing.
    \lim_{x\to\infty}f(x)=\infty, \ \lim_{x\to\ -\infty}f(x)=-\infty and f is continuous, so f has at least a real root. But f is injective (because is strictly increasing) and the root is unique.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Bryn View Post
    y=x^3+8x-19

    I know the derivative is 3x^2+8, but how does this tell me there is no turning point and therefore that the function is increasing has one root

    thanks

    Bryn
    See this thread. There is an entirly different proof there from any in this thread as yet.

    RonL
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