Question

Use Lagrange multipliers to find the point on the given plane that is closest to the following point.

x - y + z = 4; (2, 7, 7)

Answer 1

Using Lagrange multipliers, the task is to find the point on the plane x - y + z = 4 that is closest to the point (2, 7, 7) by minimizing the distance squared function subject to the given plane equation.

Step-by-step explanation:

To find the point on the plane x - y + z = 4 that is closest to the point (2, 7, 7), we can use the method of Lagrange multipliers. This method involves defining a distance function d(x,y,z) from the point to a generic point (x,y,z) on the plane and then minimizing this function subject to the constraint x - y + z = 4. The distance squared function d^2 is given by d^2 = (x-2)^2 + (y-7)^2 + (z-7)^2. To apply Lagrange multipliers, we set up the system of equations derived from the partial derivatives of L = d^2 + λ(x - y + z - 4), where λ is the Lagrange multiplier. The system to solve is then:

• 2(x - 2) + λ = 0
• 2(y - 7) - λ = 0
• 2(z - 7) + λ = 0
• x - y + z - 4 = 0

Solving this system of equations will give us the point on the plane that is closest to the point (2, 7, 7).