1. 1

2. 2 Local maximum Local minimum

3. 3 Saddle point

4. 4 Given the problem of maximizing ( or minimizing) of the objective function: Z=f(x,y ) Finding the Stationary Values solutions of the following system:

5. 1) Z=f(x,y)=x 2 +y 2 2) Z=f(x,y)=x 2 -y 2 3) Z=f(x,y)=xy 5

6. 6 The Hessian Matrix H(x 0,y 0 )>0 f xx >0 minimum H(x 0,y 0 )>0 f x x <0 maximum H(x 0,y 0 )<0 saddle

7.  The method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function:  subject to constraints: 7

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10. 10 For instance minimize the objective function Subject to the constraint:

11.  We can combine the constraint with the objective function:  Minimum in P(1/2;1/2) 11

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13.  We introduce a new variable ( λ ) called a Lagrange multiplier, and study the Lagrange function: 13

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16. 16 The point is a minimum The point is a maximum Bordered Hessian Matrix of the Second Order derivative is given by

17.  Given the problem of maximizing ( or minimizing) of the objective function with constraints 17

18.  We build a Lagrangian function :  Finding the Stationary Values: 18

19.  Second order conditions:  We must check the sign of a Bordered Hessian: 19

20.  n=2 e m=1  the Bordered Hessian Matrix of the Second Order derivative is given by  Det>0 imply Maximum  Det<0 imply Minimum 20

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22.  Case n=3 e m=1  : 22

23. 23

24.  n=3 e m=2  the matrix of the second order derivate is given by: 24

25. 25