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Math Help - Find Where f Increase, Decrease, Local Minimum, Inflection Point, concave up and down

  1. #1
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    Find Where f Increase, Decrease, Local Minimum, Inflection Point, concave up and down

    f(x) = x^3 lnx

    f'(x) = x^2 (3lnx +1)

    x= e^(-1/3)

    left side of e^(-1/3) is negative, right side is positive so

    f increase at (e^(-1/3), infinity)
    f decrease (-infinity , e^(-1/3))
    correct?

    Local Minimum:
    f(e^(-1/3)) = .123 so local minimum is:
    .123 correct?


    Inflection point:

    f'(x) = x^2 (3lnx +1)

    f "(X) = 6xlnx + 3x + 2x
    = x (6lnx + 5) = 0

    x= e^(-5/6)

    ----- e^(-5/6) ++++

    I don't know what to do from here..
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Find Where f Increase, Decrease, Local Minimum, Inflection Point, concave up and

    Quote Originally Posted by NeoSonata View Post
    f increase at (e^(-1/3), infinity)
    Right.

    f decrease (-infinity , e^(-1/3)) correct?
    Wrong, f is defined in (0,+\infty). As a consequence, it is decreasing in (0,e^{-1/3}) .
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  3. #3
    MHF Contributor Siron's Avatar
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    Re: Find Where f Increase, Decrease, Local Minimum, Inflection Point, concave up and

    Quote Originally Posted by NeoSonata View Post
    Inflection point:

    f'(x) = x^2 (3lnx +1)

    f "(X) = 6xlnx + 3x + 2x
    = x (6lnx + 5) = 0

    x= e^(-5/6)

    ----- e^(-5/6) ++++

    I don't know what to do from here..
    You have found x=e^{-\frac{5}{6}} wherefore f''(x)=0. So what's the inflection point (x and y coordinate)? ...
    You can now determine where the functions is convex or concave.
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  4. #4
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    Re: Find Where f Increase, Decrease, Local Minimum, Inflection Point, concave up and

    we have to check f '
    f ' is negative if \ln{x} < \frac{1}{3}
    gives x < e^{\frac{1}{3}
    therefore f is decreasing if x \in ( 0 , e^{\frac{1}{3} )
    and f is increasing if x \in ( e^{\frac{1}{3} , \infty)
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