The document provides examples of using Lagrange multipliers to find the extremum of a function subject to a constraint. In example 8, the critical point and extremum are found for f(x,y,z) = x + y + z with the constraint x + y + z = 1. In example 9, the critical point (0, 0, 0) is identified as minimizing the distance from any point (x,y,z) to the origin. Example 10 finds the critical point (0, √2, √2) minimizes the distance function with the constraint x^2 + y^2 + z^2 = 2.

1. TUTOR 7. LAGRANGE MULTIPLIER
8) Find the extremum of with the condition
̃ ̃ ̌ ̃ ̃ ̌
(1) + (2) + (3) =
Substitute into (1), (2) and (3).
(do as your exercise)
So, critical point
( )
Extremum of at the critical point
( ) ( ) ( )

2. 9. Distance from any point (x,y,z) to the origin:
√
(1) Into (2):
When z=0, x=0,
So, the critical point is (0, ,0)

3. 10. ,
Distance to origin :
Find x, y and z as your exercise.
Critical point: (0, √ , √ )
(
√
) (
√
)
Distance = 1 at (0, √ , √ ).

4. 11. ,
Find x,y and z. Do as your exercise.
Critical point: ( √ √ √ )
( √ √ √ ) ( √ )( √ ) ( √ )( √ ) ( √ )( √ )