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Math Help - Another optimization word prob.

  1. #1
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    Another optimization word prob.


    Explain why the minimum value of x^2+y^2 on x+y=4 occurs at the point at which a graph of x^2+y^2=Constant is tangent to the line x+y=4

    Then

    Using this answer, and implicit differentiation to find the slope of the circle, find the minimum value of x^2+y^2 such that x+y=4.
    This assignment is my Everest.
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  2. #2
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    Explain why the minimum value of x^2+y^2 on x+y=4 occurs at the point at which a graph of x^2+y^2=Constant is tangent to the line x+y=4
    Suppose x^2+y^2=k^2, this meets the line when:

    x^2+(4-x)^2=k^2,

    or:

    2x^2-8x+(16-k^2)=0

    which has roots:

    x= [8+/-sqrt(64-8(16-k^2))]/4=2 +/- sqrt(-4+k^2/2)

    Now for large k this has two solutions but as k is reduced there comes a point
    where k^2/2=4, where there is only one root, any smaller and there are no roots.

    So the minimum k for which there are roots corresponds to the line being
    a tangent.



    Then

    Using this answer, and implicit differentiation to find the slope of the circle, find the minimum value of x^2+y^2 such that x+y=4.
    This seems over complex but at a point of tangency the slope of the
    circle will be equal to that of the line (-1).

    d/dx(x^2+y^2)=2x +2y dy/dx=0,

    so:

    dy/dx=-x/y.

    The slope of the line is -1, so x=y at the point of tangenct, so:

    2x=4, so x=2, y=2 and hence the minimum of x^2+y^2=8.

    RonL
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  3. #3
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    Thank you thank you thank you thank you
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