Results 1 to 9 of 9

Math Help - Decreasing/concave upward: xlnx

  1. #1
    Banned
    Joined
    Apr 2010
    Posts
    58

    Decreasing/concave upward: xlnx

    a) On what interval is f(x) = xlnx decreasing?
    b) On what interval is f concave upward?

    My attempt:

    a) f'(x) = (x)d/dx(lnx) + (lnx)d/x(x)
    = x * 1 / x + lnx
    = lnx

    According to lnx graph, f(x) decreasing when 0 < x < 1

    b) f''(x) = 1 / x

    According to 1 / x graph f concave upward when x < 0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,576
    Thanks
    1416
    Quote Originally Posted by TsAmE View Post
    a) On what interval is f(x) = xlnx decreasing?
    b) On what interval is f concave upward?

    My attempt:

    a) f'(x) = (x)d/dx(lnx) + (lnx)d/x(x)
    = x * 1 / x + lnx
    = lnx
    x*(1/x)= 1, not 0!

    According to lnx graph, f(x) decreasing when 0 < x < 1

    b) f''(x) = 1 / x

    According to 1 / x graph f concave upward when x < 0
    The derivative of x ln(x) is 1+ ln(x), not ln(x), but the second derivative is 1/x so this is correct.

    For what values of x is 1+ ln(x)< 0?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Quote Originally Posted by TsAmE View Post
    b) On what interval is f(x)= x\cdot \ln x concave upward?

    My attempt:

    b) f''(x) = 1 / x

    According to 1 / x graph f concave upward when x < 0
    There is only one minor problem: for x<0 the function f(x) = x\cdot \ln x is not a real function, so that the concept of 'concave upward function' hasn't the usual meaning...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Apr 2010
    Posts
    58
    Quote Originally Posted by HallsofIvy View Post
    For what values of x is 1+ ln(x)< 0?
    x + ln(x) < 0 when x < 1/e, ok that makes sense, and how would you get the 0 in (0, 1/e)? I got it for my wrong answer by sketching the lnx graph and looking at it, but clearly that is wrong in this case. For b I tried making an equality 1 / x > 0 to solve for x and find when it is concave upward, but that doesnt work.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    In order to find a function 'concave upward' for x<0 we can start to the second order DE...

    y^{''} = \frac{1}{x} (1)

    The (1) is easily integrable and is...

     y(x) = x\cdot \ln |x| + c_{1}\cdot x + c_{2} (2)

    If we impose the conditions y(0)=0 and y^{'} (1)=1 we obtain c_{1}= c_{2} =0 so that the function we are loocking for is...

    y(x)= x\cdot \ln |x| (3)

    ... a 'beautiful odd function'...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Apr 2010
    Posts
    58
    Sorry but I havent done integration yet, only doing that later this year
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    The 'beautiful odd function' f(x)= x\cdot \ln |x| is illustrate there...



    It is a 'upward concave function' for x<0 , whereas x\cdot \ln x has complex value in that interval...

    The reason for which I have called this function 'beautiful' is relative to a ' still unresolved problem' the solution of which is strictly connected to it...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Feb 2010
    Posts
    148
    Thanks
    2
    Quote Originally Posted by chisigma View Post
    The 'beautiful odd function' f(x)= x\cdot \ln |x| is illustrate there...



    It is a 'upward concave function' for x<0 , whereas x\cdot \ln x has complex value in that interval...

    The reason for which I have called this function 'beautiful' is relative to a ' still unresolved problem' the solution of which is strictly connected to it...

    Kind regards

    \chi \sigma
    Isn't it an 'upward concave function' for x>0?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    According to...

    Convex function - Wikipedia, the free encyclopedia

    ... is seems that ione is right!... on effect 'upward concave' is the same as 'convex' , so that f(x)= x\cdot \ln |x| is convex for x>0 and concave for x<0 ...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: November 6th 2011, 03:39 PM
  2. Replies: 1
    Last Post: August 24th 2010, 07:06 AM
  3. Replies: 4
    Last Post: October 20th 2009, 12:12 PM
  4. Derivative of e^(xlnx)
    Posted in the Calculus Forum
    Replies: 6
    Last Post: March 30th 2008, 09:38 PM
  5. Increasing/Decreasing/Concave up/ concave down
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 16th 2008, 05:43 PM

Search Tags


/mathhelpforum @mathhelpforum