(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

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# Extrema in 3D problem

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