How to Find Extrema of Linear Functions on a Sphere?

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SUMMARY

The discussion focuses on finding the extrema of linear functions on the surface of a sphere defined by the equation x² + y² + z² = 1. The user initially attempted to use partial derivatives but encountered difficulties as none yielded zero. The solution provided involves applying the method of Lagrange multipliers, which is a crucial technique for optimizing functions subject to constraints.

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  • Understanding of multivariable calculus
  • Familiarity with Lagrange multipliers
  • Knowledge of the equation of a sphere in three-dimensional space
  • Basic skills in finding maxima and minima of functions
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  • Study the application of Lagrange multipliers in various optimization problems
  • Explore examples of finding extrema on different surfaces
  • Learn about the geometric interpretation of Lagrange multipliers
  • Review concepts of partial derivatives and their role in optimization
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Students and professionals in mathematics, particularly those studying calculus and optimization techniques, as well as anyone looking to enhance their problem-solving skills in multivariable contexts.

jschmid2
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So, I seem to be drawing some blanks - my calc III is a little rusty.

Find the max(2x+2y+z) and min(2x+y+z) on the surface x2+y2+z2=1

At first I thought I would take partial derivatives, but none of them yield 0, so that's not going to work. Any suggestions would be mighty helpful because I will be dealing with problems a little harder than this, but if I can get this fundamental figured out, it will be extremely helpful.

Thanks.
 
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Try LaGrange multipliers.
 
That is exactly what I needed. Thank you so much for the hint! :)
 

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