What is the shortest distance from the surface to origin

Question

What is the shortest distance from the surface xy+15x+z^2 to the origin?

Answer 1

Let me first just tell you the general approach to this general question:

"What is the shortest distance from the surface g(x,y,z) = 0 to the origin?"

Form the function

f(x,y,z) = x² + y² + z²

and find the minimum of f(x,y,z) subject to g(x,y,z) = 0.

The reason this will answer the given question is because f(x,y,z) is

the square of the distance between the origin and the point (x,y,z).

You solve this minimization problem using the approach of Lagrange multipliers.

Now let's go back to your specific problem.

An expression like "xy +15x +z²" does not define a surface, only an equation

defines a surface.

So let's assume that the statement means the surface "xy +15x +z² = 0".

In that case you can apply the above general approach to get the critical point (0,0,0).

But you do not need to go though the trouble, because the surface

"xy +15x +z² = 0" obviously passes through the origin, so the minimal distance is 0.