Question

Use Lagrange multipliers to find the point(s) on the sphere x 2 +y 2 +z 2 =1 that are at the greatest distance from the point (0,1,2).

Answer 1

Using Lagrange multipliers, we can find the points on the sphere x^2 + y^2 + z^2 = 1 that are furthest from (0,1,2) by maximizing the squared distance function subject to the constraint of the sphere.

Step-by-step explanation:

To find the point(s) on the sphere x2 + y2 + z2 = 1 that are at the greatest distance from the point (0,1,2), we use Lagrange multipliers to solve the optimization problem with a constraint. First, the distance squared between any point (x,y,z) on the sphere and the point (0,1,2) is D2 = x2 + (y-1)2 + (z-2)2. We want to maximize D2 under the constraint that x2 + y2 + z2 = 1.

The Lagrange function is L(x,y,z,λ) = D2 - λ(x2 + y2 + z2 - 1). The next step is to take the gradient of L and set it equal to zero: ∇L = <0,0,0>. This results in a system of equations that will give us the critical points of L. Solving these equations will reveal the points on the sphere that are furthest from (0,1,2).