Mathe III Lecture 8 Mathe III Lecture 8

2 Constrained Maximization Lagrange Multipliers At a maximum point of the original problem the derivatives of the Lagrangian vanish (w.r.t. all variables).

3 Constrained Maximization Lagrange Multipliers Intuition x y iso- f curves f(x,y) = K assume +

4 Constrained Maximization Lagrange Multipliers Intuition x y

5 Constrained Maximization Lagrange Multipliers Intuition x y

6 Constrained Maximization Lagrange Multipliers Intuition x y

7 Constrained Maximization Lagrange Multipliers Intuition x y

8 Constrained Maximization Lagrange Multipliers Intuition A stationary point of the Lagrangian

9 Constrained Maximization The general case

10 Constrained Maximization The general case differentiating w.r.t. x s, s = m+1,…,n

11 Constrained Maximization The general case

12 Constrained Maximization The general case

13 Constrained Maximization The general case

14 Constrained Maximization The general case The derivatives w.r.t. x m+1,…..x n are zero at a max (min) point.

15 Constrained Maximization The general case

16 Constrained Maximization The general case But:

17 Constrained Maximization The general case

18 Constrained Maximization The general case We need to show this for s = 1,….m

19 Constrained Maximization The general case

20 Constrained Maximization The general case

21 Constrained Maximization The general case define:

22 Constrained Maximization Interpretation of the multipliers

23 Constrained Maximization Interpretation of the multipliers But:

24 Constrained Maximization Interpretation of the multipliers

25 Constrained Maximization Interpretation of the multipliers 9