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Maximum distance

  1. Aug 8, 2008 #1
    1. The problem statement, all variables and given/known data

    Hey.
    I need to find the biggest distance between this function and the XY plot, any ideas?

    2. Relevant equations



    3. The attempt at a solution
     

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  3. Aug 8, 2008 #2

    HallsofIvy

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    You mean the xy-plane.
    The surface is given by [itex]2x^3+ 3y^2+ 2z^2+ 2xz= 6[/itex] and the distance from any point (x, y, z) to (x, y, 0) (the xy-plane) is just z. To minimize of maximize that, look at the derivatives of z with respect to x and y. Using the chain rule to differentiate with respect to x, [itex]6x^2+ 4zz_x+ 2z+ 2xz_x= 0[/itex] so [itex]z_x= -(6x^2+ 2z)/(4z+ 2x)= -(3x^2+ z)/(2z+x)=0[/itex] and differentiating with respect to y, [itex]6y+ 2zz_y+ 2xz_y= 0[/itex] so [itex]z_y= -6y/(2z+ 2x)= -3y/(z+ x)= 0[/itex]. Find values of of x, y, z that satisfy those as well as [itex]2x^3+ 3y^2+ 2z^2+ 2xz= 6[/itex].
     
  4. Aug 8, 2008 #3
    10x a lot
     
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