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

[porroadventum]

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...

 

[tiny-tim]

welcome to pf!

hi porroadventum! welcome to pf!

… 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 )

you are given an interval …

it's the whole plane! ​

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"

 

[porroadventum]

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...)

 

[porroadventum]

*x³-x not ^{x^3}-x, woops!

 

[tiny-tim]

… 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!

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

 

[porroadventum]

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