# Maximum distance

1. Aug 8, 2008

### asi123

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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2. Aug 8, 2008

### HallsofIvy

Staff Emeritus
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 $2x^3+ 3y^2+ 2z^2+ 2xz= 6$.

3. Aug 8, 2008

### asi123

10x a lot

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