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Math Help - intervals using 1st derivative test??

  1. #1
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    intervals using 1st derivative test??

    Find the intervals of inc and dec & local max and min of h(x)=x^3/2-256x? using 1st derivative test.?


    thanks...this one is really getting me!!!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by monolith View Post
    Find the intervals of inc and dec & local max and min of h(x)=x^3/2-256x? using 1st derivative test.?


    thanks...this one is really getting me!!!
    Is it h(x)=x^{\frac{3}{2}}-256x or h(x)=\tfrac{1}{2}x^3-256x??

    --Chris
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  3. #3
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    sorry

    i guess i wasnt using proper scripting...it is the first one you have written

    h(x)=x^{\frac{3}{2}}-256x
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by monolith View Post
    i guess i wasnt using proper scripting...it is the first one you have written

    h(x)=x^{\frac{3}{2}}-256x
    First find h'(x):

    h'(x)=\tfrac{3}{2}x^{\frac{1}{2}}-256

    Now set this equal to zero:

    \tfrac{3}{2}x^{\frac{1}{2}}-256=0\implies x^{\frac{1}{2}}=\tfrac{512}{3}\implies x=\tfrac{262144}{9}\approx 29127.1

    This is the only critical point. So we investigate the behavior of h'(x) over the intervals \left(0,\tfrac{262144}{9}\right)\text{ and }\left(\tfrac{262144}{9},\infty\right)

    Pick a point in these intervals and plug them into h'(x), for simplicity, I'll pick 1 and 360,000 (the last one seems large, but when we take the square root of it, we get a nice number to work with...no decimals involved).

    h'(1)=\tfrac{3}{2}(1)^{\frac{1}{2}}-256=\color{red}\boxed{-254.5}

    It is decreasing here.

    h'(360,000)=\tfrac{3}{2}(360,000)^{\frac{1}{2}}-256=\color{red}\boxed{644}

    It is increasing here.

    So to answer the first part of the question, the interval where its:

    increasing -- \left(\tfrac{262144}{9},\infty\right)
    decreasing -- \left(0,\tfrac{262144}{9}\right)

    where \tfrac{262144}{9}\approx 29127.111

    Now, take note that a maximum occurs when a function goes from increasing to decreasing at a critical point, and a minimum occurs when a function goes from decreasing to increasing at a critical point. Which one do you think it is? What is the coordinates of the maximum/minimum? I leave that for you to do.

    I hope this makes sense!

    --Chris
    Last edited by Chris L T521; August 13th 2008 at 10:52 PM. Reason: clarification
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