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Math Help - Cube root

  1. #1
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    Cube root

    f (x) = (x − a)3 + (x − b)3 + (x − c)3 , a < b < c.
    The number of real roots of f (x) = 0 is
    (A) 3; (B) 2; (C) 1; (D) 0.

    Please guide me with this question..
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  2. #2
    MHF Contributor red_dog's Avatar
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    \lim_{x\to -\infty}=-\infty, \ \lim_{x\to\infty}=\infty\Rightarrow f has at least one real root.

    f'(x)=3(x-a)^2+3(x-b)^2+3(x-c)^2>0, \forall x\in\mathbf{R}\Rightarrow f is strictly increasing, so f is injective. Therefore, f has only one real root.
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