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Math Help - Concavity & Inflection point(s)

  1. #1
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    Unhappy Concavity & Inflection point(s)

    This is the problem:

    I have to find:
    A) critical values = 0
    B) where f(x) is increasing = (0,inf)
    C) where f(x) is decreasing = (-inf,0)
    D) local maxima at x = none
    E) local minima at x = 0
    F) concave up =
    G) concave down =
    H) inflection point(s) at x =
    I) horizontal asymptote(s) y = 3
    J) vertical asymptote(s) x = none

    I was able to figure out everything excluding F, G, and H -as you may have noticed. I know that I have to find the second derivative of the original function however when I tried this was what I got:
    \frac{-150(x^4-2x^3+50x^2-50x+625)}{(x^2+25)^2}
    Would someone please help me?
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  2. #2
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    Usually, to find concavity, you must set the second derivative equal to zero. Use those x values to find where it is positive, where it is negative. Positive means concave up, negative means concave down.
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  3. #3
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    i set everything equal to zero and i end up with +/-\sqrt{-25} and \frac{2+/-\sqrt{-96}}{2} and i really dont think they're looking for complex roots
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  4. #4
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    Your second derivative is not correct. Try reworking it.

    But first, did you get the first derivative correct?

    f'(x) = \frac{150x}{(x^2+25)^2}
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  5. #5
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    f''(x) = \frac{(x^2+25)^2(150) - 2(x^2+25)(2x)(150x)}{(x^2+25)^4}

    Factoring out a (x^2+25) first will make the algebra easier:

    f''(x) = \frac{(x^2+25)(150) - 2(2x)(150x)}{(x^2+25)^3}
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  6. #6
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    [drumist] yup that was what i got for the first derivative.where i messed up was that i didn't take the derivative of x^2 for g prime.i fixed my error and found the inflection point to be +/-\frac{5\sqrt3}{3} and it is correct.i plugged in points near the interval and found that it is concave up at (-\inf,\frac{5\sqrt3}{3}) and concave down at (\frac{5\sqrt3}{3},\inf) however it said i was wrong.i dont understand how its wrong
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  7. #7
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    Since you have two inflection points, you have three intervals to consider!

    \left(-\infty,-\tfrac{5\sqrt{3}}{3}\right)

    \left(-\tfrac{5\sqrt{3}}{3},\tfrac{5\sqrt{3}}{3}\right)

    \left(\tfrac{5\sqrt{3}}{3},\infty\right)
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  8. #8
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    0o0k.i got it now.thank you!
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