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Math Help - Finding K such that this function has critical points

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    Finding K such that this function has critical points

    Determine the conditions on parameter k such that the function \displaystyle f(x)=\frac{2x+4}{x^2-k^2} will have critical points.

    Critical points are defined as where f'(x)=0 or when f'(x) does not exist. I was thinking it would be k>0 but the back of the book has the answer as k being between -2 and 2.

    Was my reasoning correct or am I missing something? I'm not sure why k has to be between -2 and 2.
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    Quote Originally Posted by youngb11 View Post
    Determine the conditions on parameter k such that the function \displaystyle f(x)=\frac{2x+4}{x^2-k^2} will have critical points.

    Critical points are defined as where f'(x)=0 or when f'(x) does not exist. I was thinking it would be k>0 but the back of the book has the answer as k being between -2 and 2.

    Was my reasoning correct or am I missing something? I'm not sure why k has to be between -2 and 2.
    what did you get for a derivative?
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    Quote Originally Posted by skeeter View Post
    what did you get for a derivative?
    Is this correct: \displaystyle f'(x)=\frac{-2(x^2+4x+k^2)}{(x^2-k^2)^2} ?

    Just looking at the denominator, wouldn't a positive k value produce points where the slope doesn't exist? Unless they aren't including asymptotes. In that case, how would I figure out where the slope equaled zero?
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    Quote Originally Posted by youngb11 View Post
    Is this correct: \displaystyle f'(x)=\frac{-2(x^2+4x+k^2)}{(x^2-k^2)^2} ?

    Just looking at the denominator, wouldn't a positive k value produce points where the slope doesn't exist? Unless they aren't including asymptotes. In that case, how would I figure out where the slope equaled zero?
    the derivative is correct.

    note that critical values occur at x-values where the function is defined. f(x) is undefined when x = \pm k , so there are no values of x where f(x) is defined and f'(x) is undefined.

    that leaves the critical values where f'(x) = 0

    x^2 + 4x + k^2 = 0

    using the quadratic formula ...

    x = \dfrac{-4 \pm \sqrt{4^2 - 4k^2}}{2}

    for critical values to exist, the discriminant, (b^2-4ac) \ge 0

    4^2 - 4k^2 \ge 0

    solve this inequality for k
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    Thanks a lot for the help! I was somewhat on the right track, but was just a little confused. Thanks again!
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