Question

Use Lagrange multipliers to find the points on the given cone that are closest to the point (16, 4, 0).

Answer 1

To find the points on the cone that are closest to the given point using Lagrange multipliers, set up the Lagrangian function and find the critical points.

Step-by-step explanation:

To find the points on the cone that are closest to the given point (16, 4, 0) using Lagrange multipliers, we need to formulate the problem as an optimization problem. Let's define the distance between the point (x, y, z) on the cone and (16, 4, 0) as the square root of
`(x-16)^2 + (y-4)^2 + z^2`. The equation of the cone is
`x^2 + y^2 = z^2`. We want to minimize the distance function subject to the constraint equation of the cone.

Using Lagrange multipliers, we can set up the Lagrangian function
`L = (x-16)^2 + (y-4)^2 + z^2 + λ(x^2 + y^2 - z^2)`, where λ is the Lagrange multiplier. We need to find the critical points of L by solving the system of equations formed by taking partial derivatives of L with respect to x, y, z, and λ and setting them equal to zero.

After obtaining the critical points, we can calculate the distance between each critical point and (16, 4, 0). The critical point with the smallest distance will be the point on the cone that is closest to (16, 4, 0).