Question

Find the maximum value of the function f(x,y,z)=x+2y+7z on the curve of intersection of the plane x−y+z=1 and the cylinder x² +y² =1.

Answer 1

To find the maximum of the function f(x,y,z) on the intersection of the plane and cylinder, one should use the Lagrange multipliers method to solve the system of gradients related to f and the constraints, and then evaluate the maximum.

Step-by-step explanation:

To find the maximum value of the function f(x,y,z)=x+2y+7z on the curve of intersection of the plane x-y+z=1 and the cylinder x²+y²=1, we need to apply the method of Lagrange multipliers.

First, we set up two constraint equations corresponding to the given surfaces:

1. The plane: x - y + z = 1
2. The cylinder: x² + y² = 1

We then introduce two Lagrange multipliers and obtain the system of equations through the gradients of the function f(x, y, z) and the constraints.

We solve this system to find the points (x, y, z) that could potentially give the maximum value. Finally, we evaluate the function f at these points to determine which one provides the maximum value of f on the intersection curve.