Proving a local minimum is a global minimum.Let f ( x , y ) = x y + 50 x + 20 y , Find the global minimum / maximum of the function for x &gt; 0 , y &gt; 0Clearly the function has no global maximum since f is not bounded. I have found that the point ( 5 , 2 ) is a local minimum of f. It seems pretty obvious that this point is a global minimum, but I'm struggling with a formal proof.

Question

Proving a local minimum is a global minimum.Let   f  (  x  ,  y  )  =  x  y  +      50    x    +      20    y  , Find the global minimum / maximum of the function for   x  &amp;gt;  0  ,  y  &amp;gt;  0Clearly the function has no global maximum since   f is not bounded. I have found that the point   (  5  ,  2  ) is a local minimum of   f. It seems pretty obvious that this point is a global minimum, but I&#039;m struggling with a formal proof.

Julianne Mccoy · Accepted Answer

By AM-GM  f  (  x  ,  y  )  ≥  3            x      y      ⋅              50        x            ⋅              20        y              3    =  30.The equality occurs for  x  y  =      50    x    =      20    y    =  10  ,id est, for   (  x  ,  y  )  =  (  5  ,  2  ), which says that   30 is a minimal value.The maximum does not exist. Try   x  →      0    +

Answered question

Answer & Explanation

New Questions in Pre-Algebra