Proving Global Max/Min for Function w/ 2 Variables: Advice Needed

In summary, the conversation involves a person asking for help with a question involving a function with 2 variables and the difficulty of proving that it does not have a global maximum or minimum. The general strategy is to consider the turning points and the whole of the boundary. The critical points of the function are (-1/3, 1/3) and (1,-1), classified as a saddle point and local minimum, respectively. To show that the function does not have a global maximum or minimum, the person must consider the values of the function on the whole plane, including the boundary "at infinity." By setting y=0, it is possible to prove that the function reaches both +∞ and -∞ "at infinity."
  • #1
porroadventum
34
0
I am struggling with this question which appears in every adv. calculus exam paper I practise and would love some help or advice on how to even approach it! I have no trouble getting the extreme points and determining whether they are local minimum, local maximum or saddle points, but proving that a function with 2 variables does not achieve a global max or min is proving very difficult. Here is an example of a question:

1. Let f(x,y)=y^2+2xy+x^3-x. Find the critical points of f and classify each of them as a local maximum, a local minimum or a saddle point.

(The answers I have come up with for this part are : (-1/3, 1/3) is a saddle point and (1,-1) is a local minimum.

2. Consider the values of f on the x- axis, or otherwise, to show that f has neither a global maximum nor a global minimum.

I don't know what to do here, especially since I am not given an interval...
 
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  • #2
welcome to pf!

hi porroadventum! welcome to pf! :smile:
porroadventum said:
… proving that a function with 2 variables does not achieve a global max or min is proving very difficult.

the general strategy is to consider the turning points and the whole of the boundary

if it has a maximum or minimum, it must either be at a local maximum or minimum, or it must be on the boundary
1. Let f(x,y)=y2+2xy+x3-x. Find the critical points of f and classify each of them as a local maximum, a local minimum or a saddle point.

(The answers I have come up with for this part are : (-1/3, 1/3) is a saddle point and (1,-1) is a local minimum.

2. Consider the values of f on the x- axis, or otherwise, to show that f has neither a global maximum nor a global minimum.

I don't know what to do here, especially since I am not given an interval...

(try using the X2 button just above the Reply box :wink:)

you are given an interval …

it's the whole plane! :biggrin:

you have to consider the boundary, which is "at infinity" in every direction

in this case, if you follow the hint and put y = 0, you should easily prove that it reaches both +∞ and -∞ "at infinity" :wink:
 
  • #3


Sorry I can be a bit dim sometimes, so to make sure I've understood:-

BY letting y=0 I get the function x^3-x and then I calculate the limit of the function as x approaches +∞ (=+∞) and -∞ (=-∞). If the limits were to equal a real number, would this mean that this real number is the global max/ min?

(I'm really sorry if I've gotten it completely wrong...)
 
  • #4


*x3-x not x^3-x, woops!
 
  • #5
porroadventum said:
… If the limits were to equal a real number, would this mean that this real number is the global max/ min?

it could be, but you would have to compare that value with the values at the local maxima and minima … to see who wins! :biggrin:

ohh, and of course with the values at every other direction "at infinity" (not just the x and y axes :wink:)
 
  • #6


OK, thank you so much for your help! I am so grateful!
 

What is the definition of a global max/min for a function with 2 variables?

A global max/min for a function with 2 variables is the highest/lowest value that the function can take on within its entire domain.

How do I know if a function has a global max/min?

A function with 2 variables has a global max/min if it is continuous and defined on a closed and bounded region.

What is the process for proving a global max/min for a function with 2 variables?

The process for proving a global max/min for a function with 2 variables involves taking the partial derivatives of the function with respect to each variable, setting them equal to 0, and solving for the critical points. Then, the points must be evaluated to determine if they are maximum or minimum values.

Can a function have more than one global max/min?

Yes, a function with 2 variables can have multiple global max/min values. These values can occur at different points or can occur at the same point with different values for each variable.

What are some common mistakes to avoid when proving global max/min for a function with 2 variables?

Some common mistakes to avoid when proving global max/min for a function with 2 variables include not considering all critical points, not properly evaluating the points, and not considering the boundaries of the region. It is also important to check for saddle points, which can occur at critical points where the function has neither a maximum nor minimum value.

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