Max Distance: Find Largest Difference Between Function & XY Plot

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SUMMARY

The discussion focuses on finding the maximum distance between the surface defined by the equation 2x³ + 3y² + 2z² + 2xz = 6 and the XY plane. The distance from any point (x, y, z) to the XY plane is represented by the variable z. To find the extrema of this distance, participants suggest using derivatives of z with respect to x and y, applying the chain rule, and solving the resulting equations to find critical points that satisfy the original surface equation.

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Homework Statement



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

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The Attempt at a Solution

 

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asi123 said:

Homework Statement



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

10x a lot
 

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