How to Find Extrema of Linear Functions on a Sphere?

In summary, the conversation discusses finding the maximum and minimum values of an expression on a given surface. The individual initially considers using partial derivatives but realizes it is not possible. They then receive a helpful suggestion to use LaGrange multipliers.
  • #1
jschmid2
6
0
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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  • #2
Try LaGrange multipliers.
 
  • #3
That is exactly what I needed. Thank you so much for the hint! :)
 

FAQ: How to Find Extrema of Linear Functions on a Sphere?

What is the definition of a min/max of a function over a sphere?

The min/max of a function over a sphere refers to the smallest or largest value that the function can take on when evaluated at any point on the surface of a sphere.

How is the min/max of a function over a sphere calculated?

The min/max of a function over a sphere is typically calculated using the method of Lagrange multipliers, which involves finding the critical points of the function subject to the constraint that the point lies on the surface of the sphere.

What is the significance of finding the min/max of a function over a sphere?

Finding the min/max of a function over a sphere can provide important insights into the behavior of the function and can help in solving optimization problems that involve spherical constraints.

Can a function have multiple min/max values over a sphere?

Yes, a function can have multiple min/max values over a sphere, depending on the complexity of the function and the shape of the sphere. In some cases, there may be no min/max values at all.

How does the radius of the sphere affect the min/max of a function over a sphere?

The radius of the sphere can significantly impact the min/max of a function over a sphere. As the radius increases, the surface area of the sphere also increases, potentially leading to more critical points and a greater range of possible min/max values.

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