Proving a local minimum is a global minimum. Let f(x,y)=xy+ (50)/(x)+(20)/(y), Find the global minimum / maximum of the function for x>0,y>0. Clearly 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.

kennadiceKesezt 2022-09-26 Answered
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 > 0 , y > 0
Clearly 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.
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Answers (1)

Julianne Mccoy
Answered 2022-09-27 Author has 10 answers
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 +
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