Originally Posted by

**woody198403** I have this question that I am lost on:

Use lagrange multipliers to find the points on the surface z^2 - xy = 49

that are closest to the origin.

This is my thinking so far:

d = sqrt ((x-0)^2 + (y-0)^2 + (z-0)^2

or

d^2 = (x-0)^2 + (y-0)^2 + (z-0)^2

so

del f = 2(x-0)i + 2(y-0)j + 2(z-0)k

The constraint is

g(x,y,z) = z^2 - xy = 49

so lambda del g(x,y,z) = lambda(k^2 - jk)

so

2(x-0)i + 2(y-0)j + 2(z-0)k = lambda(k^2 - jk), which leads to

2(z - 0) = lambda^2 and 2(y-0)*2(x-0) = lambda^2

therefore,

lambda = sqrt(2(z-0)) = sqrt(2(y-0)*2(x-0))

so sqrt(2*z) = sqrt(4*x*y)

so 2*z = 4*x*y

Im not even sure if this is correct, but if it is where do I go from here?

I have made many attempts, and with one of those I came up with 3 points:

x=7, y =7

x=1 , y=49

x=49, x=1

but again, I dont think this is correct.