(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the points on the level surface xy^{2}z^{4}=1 that are closest to the origin.

2. Relevant equations

Lagrange's method for finding extrema

3. 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^{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?

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# Homework Help: Lagrange multiplier question

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