Max Distance: Find Largest Difference Between Function & XY Plot

[asi123]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

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 $2x^3+ 3y^2+ 2z^2+ 2xz= 6$ 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, $6x^2+ 4zz_x+ 2z+ 2xz_x= 0$ so $z_x= -(6x^2+ 2z)/(4z+ 2x)= -(3x^2+ z)/(2z+x)=0$ and differentiating with respect to y, $6y+ 2zz_y+ 2xz_y= 0$ so $z_y= -6y/(2z+ 2x)= -3y/(z+ x)= 0$. Find values of of x, y, z that satisfy those as well as $2x^3+ 3y^2+ 2z^2+ 2xz= 6$.

 

[asi123]

You mean the xy-plane.
The surface is given by $2x^3+ 3y^2+ 2z^2+ 2xz= 6$ 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, $6x^2+ 4zz_x+ 2z+ 2xz_x= 0$ so $z_x= -(6x^2+ 2z)/(4z+ 2x)= -(3x^2+ z)/(2z+x)=0$ and differentiating with respect to y, $6y+ 2zz_y+ 2xz_y= 0$ so $z_y= -6y/(2z+ 2x)= -3y/(z+ x)= 0$. Find values of of x, y, z that satisfy those as well as $2x^3+ 3y^2+ 2z^2+ 2xz= 6$.

10x a lot