MTH3101 Tutor 7 lagrange multiplier
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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.
MTH3101 Tutor 7 lagrange multiplier
- 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: ( √ √ √ ) ( √ √ √ ) ( √ )( √ ) ( √ )( √ ) ( √ )( √ )