Use the Lagrange Method to find the points on the surface that are closest to th

Question

Use the Lagrange Method to find the points on the surface that are closest to the origin.

(Hint: consider a point (x,y,z), it has distance ? from the origin. You want to minimize this distance subject to the constraint that this point is a point on the given surface, i.e. satisfies the equation . Here is a useful Trick when optimizing distances: Instead of minimizing the distance function, you can minimize, instead, the square of the distance (= ) that makes the partial derivatives easier and is as good because, the distance is minimal exactly when the distance ^2 is minimal among all the distances ^2

Explanation / Answer

The problem is

min x^2+y^2+z^2
subject to z = f(x,y)

The function to optimize is L(x,y,z,a) = x^2+y^2+z^2 +a(z-f(x,y))

dL/dx = 2x -afx = 0 where fx denotes df/dx
dL/dy = 2y-afy = 0
dL/dz = 2z+a = 0
dL/da = z-f(x,y)=0

for instance if z = xy+1

fx = y

fy=x

So we have the system :

2x-ay=0,

2y-ax=0

2z+a = 0

z =xy+1

x+yz=0

y+xz=0

=> x -xz^2 = 0

=> x(1-z^2)=0

If x= 0, then y=0 , z=1 and a=-2 < 0, so we have the point (0,0,1)

If x!=0 then z^2=1, so z=+-1 and we have the systems:

x+y=0 or x-y=0, so x=y and z=x^2+1 and x^2+x^2+(x^2+1)^2 doesn't have global minimum.

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