Extrema in 3D Problem: Finding Highest and Lowest Points on a Defined Surface

[namu]

Homework Statement

Consider the surface defined by

F(x,y,z)=1/2(x+y)^2+(y+z)^2+(x+z)^2=9

(a) Evaluate grad(F)

My Solution:
grad(F)=(3x+y+2z, x+3y+2z, 2x+2y+4z)

(b) Find the highest and lowest points on the surface (i.e. the points where z attains a maximum or minimum).

Problem:
So I can solve for z=g(x,y) using the quadratic formula and get a nasty expression (see attached) and
then go about finding the extrema, but this is ugly.

If I set each component of the gradient to zero, then the only solution is (x,y,z)=(0,0,0) which is NOT an
extrema (see attached image), rather there are two extrema.

How do I do this the "easy" way?

(c) The surface is illuminated from far above by light rays that are directed parallel to the z-axis. Find the
shape of its shadow in the plane below the surface parallel to the (x,y) coordinate plane.

Problem:

So now this is projecting the surface onto the (x,y)-plane. I have no idea how to do this.

Please help.

 

[Ray Vickson]

Homework Statement

Consider the surface defined by

F(x,y,z)=1/2(x+y)^2+(y+z)^2+(x+z)^2=9

(a) Evaluate grad(F)

My Solution:
grad(F)=(3x+y+2z, x+3y+2z, 2x+2y+4z)

(b) Find the highest and lowest points on the surface (i.e. the points where z attains a maximum or minimum).

Problem:
So I can solve for z=g(x,y) using the quadratic formula and get a nasty expression (see attached) and
then go about finding the extrema, but this is ugly.

If I set each component of the gradient to zero, then the only solution is (x,y,z)=(0,0,0) which is NOT an
extrema (see attached image), rather there are two extrema.

How do I do this the "easy" way?

(c) The surface is illuminated from far above by light rays that are directed parallel to the z-axis. Find the
shape of its shadow in the plane below the surface parallel to the (x,y) coordinate plane.

Problem:

So now this is projecting the surface onto the (x,y)-plane. I have no idea how to do this.

Please help.

Use the equation F(x,y,z) = 0 to evaluate $z_x = \partial z / \partial x$ and $z_y = \partial z / \partial y$ at a given point $(x_0,y_0,z_0)$ on the surface; that is, you need to figure out $\Delta z$ such that
$$F(x_0 + \Delta x, y_0 + \Delta y, z_0 + \Delta z) = 0.$$ You don't need the exact$\Delta z$; you just need the ``first-order'' expression that is linear in $\Delta x \mbox{ and } \Delta y.$ Then you need $z_z = 0$ and $z_y = 0.$

RGV