Find global extrema of f ( x , y ) = x 2 </msup>

preityk7t 2022-06-20 Answered
Find global extrema of f ( x , y ) = x 2 2 + 3 y 2 on the set M ( g ) := { ( x , y ) R 2 g ( x , y ) = 0 } where g ( x , y ) = x 2 + y 4 25.

M ( g ) is compact and f continuous so we know that there must exists a global maximum and global minimum. The conditions of the Lagrange multiplier methods are satisfied and if we solve the equations that result from the Lagrange multiplier method we get the following points:
( 0 , 5 ) , ( 0 , 5 ) , ( 5 , 0 ) , ( 5 , 0 ) , ( 4 , 3 ) , ( 4 , 3 ) , ( 4 , 3 ) , ( 4 , 3 ) .
and
f ( 0 , 5 ) = 15 , f ( 0 , 5 ) = 15 , f ( 5 , 0 ) = 25 2 , f ( 5 , 0 ) = 25 2 , f ( 4 , 3 ) = 17 , f ( 4 , 3 ) = 17 , f ( 4 , 3 ) = 17 , f ( 4 , 3 ) = 17.
So far so good.

However our sample solution says that "from the above values we see that 25 2 is the global minimum and 17 the global maximum. "

As far as I have understood the Lagrange multiplier method it only delivers a necessary but not sufficient condition. So we don't know if one of the three points 25 2 , 15 , 17 is a saddle point. To make sure that the points are indeed extrema we have to resort to another method (e.g. plug in the condition into f).

Am I am right or is there something I don't see or didn't understand correctly?
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Answers (2)

trajeronls
Answered 2022-06-21 Author has 21 answers
You are misunderstanding how a necessary condition works, I think.

In this problem, once you have found the set of ordered pairs from the method of Lagrange multipliers, you can be sure that the global maximizer and minimizer are in this set (if they exist). This is because being in this set is a necessary condition for being the global maximizer or minimizer. This condition isn't sufficient for being a global maximizer or minimizer - this means that it is not true that every ordered pair in the set corresponds to a global maximizer or minimizer.

Since we know that the global maximizer and minimizer are in this set of ordered pairs, we can simply plug all these points into the function and see what values we get. Whichever one gives the biggest (resp. smallest) value corresponds to the global maximizer (resp. minimizer).
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cazinskup3
Answered 2022-06-22 Author has 6 answers
The Lagrange multiplier method says that IF you are at a extreme value, it must be at a critical point. So that by itself does not guarantee they exist.
However, as you noted, the domain is compact and the function is continuous, so the extreme value theorem guarantees existence. So once you know they exist and they must be at one of the multiplier points, calculating the values there is sufficient.
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