Results 1 to 2 of 2

Math Help - real numbers

  1. #1
    Member
    Joined
    Nov 2006
    Posts
    152

    real numbers

    a, b, c, d, e are real numbers such that


    a + b + c + d + e = 8
    a^2 + b^2 + c^2 + d^2 + e^2 = 16.


    What is the largest possible value of e?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member JaneBennet's Avatar
    Joined
    Dec 2007
    Posts
    293
    Let’s suppose a, b, c, d are non-negative. (Obviously e will have to be positive if we want to maximize it.)

    Applying AM–GM to the first equation gives

    8-e=4\left(\frac{a+b+c+e}{4}\right)\ge\sqrt[4]{abcd}

    \therefore\ e\le8-4\sqrt[4]{abcd}

    and e is maxed when a=b=c=d.

    This is consistent with the application of AM–GM to the second equation, when we get

    e^2\le16-4\sqrt[4]{a^2b^2c^2d^2}=16-4d^2 when a^2=b^2=c^2=d^2.

    (Clearly, for positive e, e is maxed if and only e^2 is maxed.)

    Solving those two equations in e and d gives e=\frac{16}{5} as the maximum value for a,b,c,d\ge0.

    If some or all of a, b, c, d are negative, some other method may have to be tried.
    Last edited by JaneBennet; July 19th 2008 at 06:18 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Set of 13 real numbers
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: July 5th 2011, 09:37 AM
  2. Replies: 1
    Last Post: September 27th 2010, 04:14 PM
  3. For which real numbers
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 14th 2008, 04:25 PM
  4. Real Numbers - Real Anaylsis
    Posted in the Calculus Forum
    Replies: 5
    Last Post: September 3rd 2008, 11:54 AM
  5. For how many real numbers x does e^x=ln|x|?
    Posted in the Calculus Forum
    Replies: 12
    Last Post: August 29th 2008, 08:53 AM

Search Tags


/mathhelpforum @mathhelpforum