I am maximizing f ( x , y ) = − xgiven the constraint g ( x , y ) = x 2 − y 2 = 0To satisfy the non degenerate constraint qualification I have: D g ( x , y ) = [ 2 x − 2 y ]and the set of ( x , y ) that satisfy it is having x = y.However on setting up the Lagrange multiplier: L ( x , y , λ ) = − x + λ ( x 2 − y 2 )and getting the first order conditions: L x = − 1 + 2 λ x = 0 and L y = − 2 λ y = 0 L λ = x 2 − y 2 = 0I have a contradiction since for x = yThe first equation will give: − 2 λ x = − 1The second however shows: − 2 λ x = 0Is there anywhere I have gotten wrong here?

Question

I am maximizing  f  (  x  ,  y  )  =  −  xgiven the constraint  g  (  x  ,  y  )  =      x    2    −      y    2    =  0To satisfy the non degenerate constraint qualification I have:  D  g  (  x  ,  y  )  =  [  2  x    −  2  y  ]and the set of   (  x  ,  y  ) that satisfy it is having   x  =  y.However on setting up the Lagrange multiplier:  L  (  x  ,  y  ,  λ  )  =  −  x  +  λ  (      x    2    −      y    2    )and getting the first order conditions:      L    x    =  −  1  +  2  λ  x  =  0 and      L    y    =  −  2  λ  y  =  0      L    λ    =      x    2    −      y    2    =  0I have a contradiction since for   x  =  yThe first equation will give:  −  2  λ  x  =  −  1The second however shows:  −  2  λ  x  =  0Is there anywhere I have gotten wrong here?

g2joey15 · Accepted Answer

L    λ    =      x    2    +      y    2    =  0  is missing and you got       L    x    =  −  1  +  2  x  λ  =  0 and       L    y    =  2  y  λ  =  0 in this case you will get   −  1  +  2  λ  x  =  0 (1)  −  2  λ  y  =  0 (2)      x    2    −      y    2    =  0 (3)your system has no real solutions

I am maximizing f ( x , y ) = − x given the con

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