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

Find the maximum of

[tex]\frac{x+2y+3z}{\sqrt{x^2+y^2+z^2}}[/tex]

as (x,y,z) varies among nonzero points in [tex]R^{3}[/tex]

2. Relevant equations

I'm not sure. The section in which this problem lies in talks about scalar products, norms, distances of vectors, and orthognality. However, I fail to see how that helps to find the maximum value.

3. The attempt at a solution

1. I set [tex]f(x,y,z) = \frac{x+2y+3z}{\sqrt{x^2+y^2+z^2}}[/tex] and find the extreme values at(x,y,z) where [tex]x \neq 0, y \neq 0, z \neq 0[/tex] since the problem states that it varies among nonzero points.

2. I find the parital derivaties at (x,y,z).

denom = [tex](x^2+y^2+z^2)^{3/2}[/tex]

a. [tex]f_{x}(x,y,z)= \frac{y^2-2xy+z^2-3xz}{denom}[/tex]

b. [tex]f_{y}(x,y,z)= \frac{2x^2-yx+2z^2-3yz}{denom} [/tex]

c. [tex]f_{z}(x,y,z)= \frac{3x^2-xz+3y^2-2yz}{denom}[/tex]

3. I set up a systems of equations

a. [tex]3x^2-zx+3y^2-2yz=0[/tex]

b. [tex]2x^2-yx+2z^2-3yz=0[/tex]

c. [tex]y^2-2xy+z^2-3xz = 0[/tex]

At this point, I'm stuck. Is there a different approach or am I completely off track? Thank you.

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# Homework Help: Euclidean Space - Maximum Value

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