Maximization problem with inequalities restrictionFunction g ( x , y , z ), and x + y + z = 1 , x ≥ 0 , y ≥ 0 , z ≥ 0. Now I want to maximize g.If I ignore the inequalities, then I can use lagrangian and can solve this thing for maximum. But I am not sure how to incorporate the inequalities in the system of equations.

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Maximization problem with inequalities restrictionFunction   g  (  x  ,  y  ,  z  ), and   x  +  y  +  z  =  1  ,  x  ≥  0  ,  y  ≥  0  ,  z  ≥  0. Now I want to maximize   g.If I ignore the inequalities, then I can use lagrangian and can solve this thing for maximum. But I am not sure how to incorporate the inequalities in the system of equations.

thatuglygirlyu · Accepted Answer

You can replace   g  (  x  ,  y  ,  z  ) with   h  (  x  ,  y  )  =  g  (  x  ,  y  ,  1  −  x  −  y  ). Now you need to find the maximum of   h on  S  =  [  0  ,  1  ]  ×  [  0  ,  1  ]Which is a little easier. You know that if   h has a maximum in the interior of   S, its partial derivatives will equal zero at that points. Then you simply check for possible maximums on the edges of   S (which is slightly annoying, since you must find the maximum of   h  (  0  ,  y  )  ,  h  (  1  ,  y  )  ,  h  (  x  ,  0  ) and   h  (  x  ,  1  ), but it is doable).

Maximization problem with inequalities restriction Function g ( x , y , z

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