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Help with finding first derivative and critical points?

  1. Nov 13, 2012 #1
    1. The problem statement, all variables and given/known data
    What are the critical points of function f(x) = 2 (x^2 + 4)^(1/2) - 4x + 24 ?


    2. Relevant equations
    When f'(x) equals 0 or is undefined, x is a critical number.


    3. The attempt at a solution

    The original function is f(x) = 2 (x^2 + 4)^(1/2) - 4x + 24 .
    I got the derivative as f'(x) = [2x / (x^2 + 4)^(1/2)] - 4 .
    What are the critical points? x = 0 is the only critical point I figure since 2x = 0, x = 0 (setting the numerator equal to zero). Any confirmation here?

    Or do I need to put everything under a common denominator and figure stuff out that way?
    i.e. 2x/[(x^2+4)^(1/2)] - 4[(x^2+4)^(1/2)]/[(x^2+4)^(1/2)]
    = 2x - 4 /[(x^2+4)^(1/2)]
    = 2x - 4 /[(x^2+4)^(1/2)] * [(x^2+4)^(1/2)]/[(x^2+4)^(1/2)]
    = 2x[(x^2+4)^(1/2)]-4(x^2+4) /x^2+4
    = 2x[(x^2+4)^(1/2)]-4x^2-16 /x^2+4
    ...But now what :S....
     
    Last edited: Nov 14, 2012
  2. jcsd
  3. Nov 13, 2012 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    The function has no critical points: it is strictly decreasing on ℝ. You can see this by writing f'(x) with a common denominator. You tried this, but you wrote it incorrectly on your second line above: you wrote
    [tex]2x - \frac{4}{(x^2+4)^{1/2}},[/tex]
    but possibly you meant
    [tex] \frac{2x-4}{(x^2+4)^{1/2}},[/tex]
    which would still be wrong. I suggest you start again, and be careful.

    RGV
     
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