The problem involves finding points on the surface defined by the equation xy2z3 = 2 that are closest to the origin. The subject area pertains to multivariable calculus, specifically optimization under constraints.
The discussion is ongoing, with various approaches being considered. Some participants question the initial assumptions and the formulation of the problem, while others provide alternative methods for tackling the optimization challenge.
There is a recognition that the surface does not include the origin, which influences the approach to finding the closest points. The discussion also reflects uncertainty about the correct method to apply, with references to both calculus and algebraic techniques.
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.komarxian said: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??
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.komarxian said:Otherwise, I am very unsure as to how to approach this problem. Should I be taking partial derivatives to fin the maximum and minimums?
Mark44 said: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.
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.)komarxian said: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?
Yes, you're right. I didn't think things all the way through before writing.phyzguy said: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##.