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Thread: Find k of f(x) giving the following criteria

  1. #1
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    Find k of f(x) giving the following criteria

    $\displaystyle f(x)=x^3-kx$, where $\displaystyle k$ can be any number.

    Find the values of $\displaystyle k$ such that $\displaystyle f$ has no critical numbers, one critical number, and two critical numbers.

    If anyone could help me find one of those, I'll see if I can figure out the rest.

    I'm assuming so far we set the derivative to $\displaystyle 0$, but I'm not sure exactly what to do.
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  2. #2
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    $\displaystyle f'(x) = 3x^2 - k = 0$

    $\displaystyle \implies 3x^2 = k$

    $\displaystyle \implies x^2 = \dfrac{k}{3}$

    $\displaystyle \implies x = \pm \sqrt{\dfrac{k}{3}}$

    There are no real solutions when k < 0.

    There is exactly one solution when k = 0.

    There are exactly two solutions when k > 0.
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