Optimize f(x,y,z) = x2 + y2 + z2 - 2x

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SUMMARY

The optimization problem involves minimizing the function f(x,y,z) = x² + y² + z² - 2x under the constraint g(x,y,z) = x² + y² + z² - 25 = 0. The method of Lagrange multipliers is applied, resulting in the Lagrangian L = f - λg. The partial derivatives yield the equations Lx = 2x - 2 - 2λx = 0, Ly = 2y - 2λy = 0, and Lz = 2z - 2λz = 0. Solving these equations alongside the constraint g(x,y,z) = 0 provides the necessary conditions for optimization.

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Optimize f(x,y,z) = x2 + y2 + z2 - 2x subject to the constraint g(x,y,z) = x2 + y2 + z2 - 25 = 0

Step 1:
L = f - [tex]\lambda[/tex]g = x2 + y2 + z2 - 2x - [tex]\lambda[/tex](x2 + y2 + z2 - 25).

Step 2:
Lx = 2x - 2 - 2[tex]\lambda[/tex]x = 0
Ly = 2y - 2[tex]\lambda[/tex]y = 0
Lz = 2z - 2[tex]\lambda[/tex]z = 0

What do I do now? Please help.
 
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I see four equations and four unknowns (don't forget that g(x,y,z)=0 gives you another equation!). Have you tried solving for those unknowns?
 

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