- #1

- 986

- 9

## Homework Statement

Find the points on the level surface xy

^{2}z

^{4}=1 that are closest to the origin.

## Homework Equations

Lagrange's method for finding extrema

## The Attempt at a Solution

If I have a level surface F(x,y,z)=c, it's points closest to the origin will be the ones in which the gradient vector points to the origin. A generic vector pointing to/from the origin is G=<x,y,z>, so F must be a scalar multiple of G.

I come up with a system of equations

ßx=y

^{2}z

^{4}

ßy=2xyz

^{4}

ßz=4x

^{2}z

^{3}

xy

^{2}z

^{4}=1.

I can first simplify a little bit.

**ßx=y**

ß=2xz

ß=4x

^{2}z^{4}ß=2xz

^{4}ß=4x

^{2}z^{2}I can set the 2

^{nd}and 3

^{rd}equations equal.

2xz

^{4}=4x

^{2}z

^{2}

----> x= z

^{2}/2

I can plug that x into the first 2 equations.

(y

^{2}z

^{4})/[z

^{2}/2]=2[z

^{2}/2]z

^{4}

----> y = +/- √(z

^{4}/2)

Plugging those into the constraint xy

^{2}z

^{4}=1

----> z=4

^{(1/10)}.

Am I right? What is the most straight-forward way of solving such a problem?