Results 1 to 5 of 5

Math Help - Problem

  1. #1
    Newbie
    Joined
    Jun 2006
    Posts
    8

    Problem

    Let a and b be positive real numbers such that

    a^2 + b^2 = 1

    Prove that
    a^4 + b^4 is greater than or equal to 0.5


    Would someone be so kind as to point me in the right direction?

    P.S. Don't give me the answer just yet. I want to see if I can do it alone. Just give me an idea of how I should go about doing this.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Quick's Avatar
    Joined
    May 2006
    From
    New England
    Posts
    1,024
    Try proving
    (1-a^2)^2-(1-b^2)^2\geq\frac{a^2+b^2}{2}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,707
    Thanks
    626
    Hello, sstr!

    I found a method . . . I'll give you the game plan.


    Let a and b be positive real numbers such that: . a^2 + b^2 \,= \,1

    Prove that: a^4 + b^4 \,\geq \,\frac{1}{2}

    We have: . b^2 \,= \,1 - a^2

    Substitute into . a^4\,+\,b^4

    We will have a function of a:\;\;f(a) \;= \;a^2 + (1 - a^2)^2

    Now determine its minimum value.

    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by sstr
    Let a and b be positive real numbers such that

    a^2 + b^2 = 1

    Prove that
    a^4 + b^4 is greater than or equal to 0.5


    Would someone be so kind as to point me in the right direction?

    P.S. Don't give me the answer just yet. I want to see if I can do it alone. Just give me an idea of how I should go about doing this.
    a^2+b^2=1,

    so:

    (a^2+b^2)^2=a^4+b^4+2a^2b^2=1

    Hence:

    <br />
a^4+b^4=1-2a^2b^2<br />

    subject to a^2+b^2=1 so:

    <br />
a^4+b^4=1-2a^2(1-a^2)<br />
.

    Now find and classify the extrema of the Right Hand Side of this last equation.

    RonL
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    You can convert,
    a^4+b^4 as,
    a^4+(b^2)^2
    Thus,
    a^4+(1-a^2)^2
    Open,
    a^4+1-2a^2+a^4
    Thus,
    2a^4-2a^2+1
    Now let, x=a^2 thus,
    2x^2-2x+1
    And all you need to do is work with a quadradic (you know the max and min point)
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum