Multivariable calculus problem

[komarxian]

Homework Statement

Find the points on the surface xy^2z^3=2 that are closest to the origin

Homework Equations

The Attempt at a Solution

x,y,z=/= 0, as when x,y,z = 0 it is untrue. Right?? Otherwise, I am very unsure as to how to approach this problem. Should I be taking partial derivatives to fin the maximum and minimums?

 

[phyzguy]

Are you familiar with Lagrange multipliers? This problem is easily solved with this method. Here is a Wikipedia link on how the method works:

https://en.wikipedia.org/wiki/Lagrange_multiplier

 

[Mark44]

Homework Statement

Find the points on the surface xy^2z^3=2 that are closest to the origin

Homework Equations

The Attempt at a Solution

x,y,z=/= 0, as when x,y,z = 0 it is untrue. Right??

The surface doesn't go through the origin, so it should be fairly obvious that (0, 0, 0) is not the point on the surface that is closest to the origin.

Otherwise, I am very unsure as to how to approach this problem. Should I be taking partial derivatives to fin the maximum and minimums?

If you're working from a textbook, there should be some examples of how to find the point or points in question. Basically, you want to find any points (x, y, z) that minimize the value of $xy^2z^3 - 2$ One approach uses partial derivatives.

 

[phyzguy]

T Basically, you want to find any points (x, y, z) that minimize the value of $xy^2z^3 - 2$ One approach uses partial derivatives.

I don't think this is true. You want to find points that minimize the value of $x^2+y^2+z^2$ subject to the constraint that $xy^2z^3 - 2 = 0$.

 

[Ray Vickson]

Homework Statement

Find the points on the surface xy^2z^3=2 that are closest to the origin

Homework Equations

The Attempt at a Solution

x,y,z=/= 0, as when x,y,z = 0 it is untrue. Right?? Otherwise, I am very unsure as to how to approach this problem. Should I be taking partial derivatives to fin the maximum and minimums?

Besides the Lagrange multiplier method suggested in #2, you could use the constraint surface to solve for one of the variables in terms of the other two; then substitute that expression into the distance function (squared) $x^2 + y^2 + z^2$. For example, it is particularly easy to use the surface equation to solve for $x = x(y,z)$ in terms of $y$ and $z$; then you end up with an unconstrained minimization of some function $F(y,z) = x(y,z)^2 + y^2 + z^2$ in $y$ and $z$ alone. You can solve that using standard partial-derivative methods. (But, if you happen to know about "geometric programming" you will realize that your function $F(y,z)$ is a three-term posynomial in two variables, so is a "zero degree-of-difficulty" Geometric programming problem which is solvable without calculus, using just simple algebra.)

 

[Mark44]

I don't think this is true. You want to find points that minimize the value of $x^2+y^2+z^2$ subject to the constraint that $xy^2z^3 - 2 = 0$.

Yes, you're right. I didn't think things all the way through before writing.