(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Determining Local Extrema with Lagrange

1. The problem statement, all variables and given/known data

Find local extram of f(x,y,z) = 8x+4y-z with constraint g(x,y,z) = x^2 + y^2 + z^2 = 9

2. Relevant equations

[tex]\nabla f(x,y,z) = \lambda g(x,y,z)[/tex]

3. The attempt at a solution

So I did the partial derivatives for F and G:

[tex]\nabla f(x,y,z) = (8,4,-1)[/tex] [tex]\nabla g(x,y,z) = (2x,2y,2x)[/tex]

Now I use Lagrange to get

[tex]8 =2 \lambda x , 4 = 2\lambda y , -1 =2 \lambda z[/tex]

Now I isolated [tex]\lambda = \frac{4}{x} = \frac{2}{y} =- \frac{1}{2z}[/tex]

Now I re-write both y and z in terms of x and I get:

[tex]y = \frac{x}{2}, z =- \frac{x}{8}[/tex]

Then I put them back in to g(x,y,z) = g(x,x/2,x/8)

and get:

[tex]g(x,x/2,x/8) = x^2 + \frac{x^2}{4} +\frac{x^2}{64} = 9 \Rightarrow \frac{64x^2 + 16x^2 + x^2}{64} = 9 \Rightarrow \frac{81x^2}{64} = 9 \Rightarrow x = \sqrt{\frac{193}{27}}[/tex]

but the book says x = 8/3

Can anyone see what mistake did I make? Thank You.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Determining Local Extrema with Lagrange

**Physics Forums | Science Articles, Homework Help, Discussion**