Find Shortest Distance from z2+3x-xy=9 to Origin

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To find the shortest distance from the surface defined by z² + 3x - xy = 9 to the origin, the gradient of the function was calculated, resulting in fx = 3 - y, fy = -x, and fz = 2z. However, the gradient approach is not suitable for determining the distance to the origin. The discussion suggests using Lagrange multipliers as a more effective method for this problem. The participant expressed confusion about the next steps after their initial calculations. Clarifying the use of Lagrange multipliers could help in solving for the shortest distance.
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Homework Statement


a) what is the shortest distance from the surface z2+3x-xy=9 to the origin?

Homework Equations


I know that when you take the gradient of an equation the gradient is perp. to the vector and gives the direction of largest rate of inc.


The Attempt at a Solution


so I took the gradient of f(x,y,z) and i got
fx=3-y
fy=-x
fz=2z
I tried looking at these and thinking of it as a parameterized line and i got an answer of (3,0,0) but I don't think that it is right..
now I'm still a little confused on what i should do next, can someone help point me in the right direction?
 
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The gradient won't help you find distance to the origin. I would recommend Lagrange multipliers here.
 

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