For example, x 2 + y 2 + 2 x y. We get y = − x as a critical point. Maximim/minimum of a critical point that is a function?

Question

For example,       x    2    +      y    2    +  2  x  y. We get   y  =  −  x as a critical point. Maximim/minimum of a critical point that is a function?

Neil Short · Accepted Answer

∇  f  (  x  ,  y  )  =  [  2  x  +  2  y  ,  2  y  +  2  x      ]    t  We set this to zero and get that indeed   y  =  −  x. And the hessian is  H  (  x  ,  y  )  =  (  2  ,  2  ;  2  ,  2  )Because the Hessian is (strictly) positive definite (has positive eigenvalues) it is the case that this function is convex, so any minimizer is a global minimizer.In particular, let&#039;s plug this back in the original problem, thenIn particular, let&#039;s plug this back in the original problem, thenSo there is no maximum but the minimum is zero for   (  x  ,  y  ) such that   (  x  ,  y  ).

For example, x^2+y^2+2xy. We get y=−x as a critical point. Maximim/minimum of a critical point that is a function?

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