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Math Help - Quadratic Functions

  1. #1
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    Quadratic Functions

    If x and y are real and x+y=1, find the minimum value of 7x^2 - 4xy + 9y^2 and the values that x and y then assume</SPAN>

    Thanks.
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    Quote Originally Posted by Lukybear View Post
    If x and y are real and x+y=1, find the minimum value of 7x^2 - 4xy + 9y^2 and the values that x and y then assume</SPAN>

    Thanks.
    Hi.

    ok you have that y=1-x and z=7x^2-4xy+9y^2.

    Now substitute the first into the second. This gives:
    z=7x^2-4x(1-x)+9(1-x)^2
    so
    z=2x^2-22x-9

    Now differentiate and equate to zero to find the turning point
    \frac{dz}{dx}=4x-22=0
    so
    x=5.5

    (you can clarify that this is a minimum by differentiating again and seeing it is positive)

    Now plug this into z=2x^2-22x-9 to get z=-69.5 as the min value.

    Now plug x=5.5 back into y=1-x to get y=-4.5

    so 5.5,-4.5 are the values of x and y where the minimum occurs.

    HTH
    Si.

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  3. #3
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    Since this was listed in "Pre-Algebra and Algebra", here's how to do it without calculus:

    Reduce to 2x^2- 22x- 9 as bananaxxx did and then complete the square.

    2x^2- 22x- 9= 2(x^2- 11x) - 9= 2(x^2- 11x+ (11/2)^2- (11/2)^2)- 9
    = 2(x^2- 11x+ 121/4)- 121/2- 9=  2(x- 11/2)^2- 139/2

    If x= 11/2= 5.5, x-11/2= 0 and so z= -139/2. If x is any other number, x- 11/2 is not 0 so its square is positive and z is larger than -139.2. -139.2 is the smallest possible value of z and it occurs when x= 5.5. Of course, as bananaxxx says, when x= 5.5, y= 1- 5.5= -4.5.
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  4. #4
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    Quote Originally Posted by bananaxxx View Post
    z=7x^2-4x(1-x)+9(1-x)^2
    so
    z=2x^2-22x-9
    No; z = 20x^2 - 22x + 9
    So x = .55, y = .45 ; min = 2.95
    Last edited by Wilmer; July 27th 2009 at 07:31 AM. Reason: typo -9 changed to +9 (thanks Skeeter)
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    Quote Originally Posted by Wilmer View Post
    No; z = 20x^2 - 22x - 9
    So x = .55, y = .45 ; min = 2.95
    Sure about that...?
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    Quote Originally Posted by HallsofIvy View Post
    Since this was listed in "Pre-Algebra and Algebra", here's how to do it without calculus:
    Good point well made.
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  7. #7
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    Quote Originally Posted by Lukybear View Post
    If x and y are real and x+y=1, find the minimum value of 7x^2 - 4xy + 9y^2 and the values that x and y then assume
    z = 7x^2 - 4x(1-x) + 9(1-x)^2

    z = 7x^2 - 4x + 4x^2 + 9(1 - 2x + x^2)

    z = 7x^2 - 4x + 4x^2 + 9 - 18x + 9x^2

    z = 20x^2 - 22x + 9

    x_{min} = -\frac{b}{2a} = \frac{22}{40} = \frac{11}{20}<br />

    y = 1 - x = \frac{9}{20}



    No; z = 20x^2 - 22x - 9
    So x = .55, y = .45 ; min = 2.95
    Wilmer's calculations are correct ... he just has a sign typo for the "9".
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  8. #8
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    Quote Originally Posted by skeeter View Post
    Wilmer's calculations are correct ... he just has a sign typo for the "9".
    We were both wrong then

    Edit: It seems the more I learn in maths...the more stupid mistakes I make. Im now a fourth year uni student and I can't multiply out brackets without a mistake! Oh dear...I blame it on the hangover.....
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  9. #9
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    Quote Originally Posted by bananaxxx View Post
    We were both wrong then
    I disagree ... a typo is not an algebra error.
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    Quote Originally Posted by skeeter View Post
    I disagree ... a typo is not an algebra error.
    Prove it.

    sigh...back to my ergodic theory...much easier than multiplying out brackets haha!
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  11. #11
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    Quote Originally Posted by bananaxxx View Post
    Prove it.
    Wilmer's minimum value calculation is correct.
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  12. #12
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    Quote Originally Posted by skeeter View Post
    Wilmer's minimum value calculation is correct.
    Lol i can see it is now ...just what he had written wasn't. And in a proof a typo is indistinguishable from an error!
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  13. #13
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    Hey Bananas, if you look closely, you'll see that my -9 was simply
    copying YOURS...I thought a university student would NEVER make
    a typo
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  14. #14
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    Quote Originally Posted by Wilmer View Post
    Hey Bananas, if you look closely, you'll see that my -9 was simply
    copying YOURS...I thought a university student would NEVER make
    a typo
    Hence why I said earlier about me making more and more simple mistakes the more maths I learn...do I sense hostility in you? And FYI university students are not machines we are human and make just as many mistakes as you retired folk.
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  15. #15
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    Quote Originally Posted by bananaxxx View Post
    > ...do I sense hostility in you?

    You shouldn't...twas a joke.

    > And FYI university students are not machines we are human...

    I see you like to joke too
    .
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