Engineering Calculus Notes 337

Engineering Calculus Notes 337 - 325 3.6 EXTREMA and the...

This preview shows page 1. Sign up to view the full content.

3.6. EXTREMA 325 and the value there is g parenleftbigg 1 2 parenrightbigg = f parenleftbigg 1 2 , 1 2 parenrightbigg = 1 2 . But we also need to consider what happens when bardbl ( x,y ) bardbl→∞ in our set. It is easy to see that for any point ( x,y ), f ( x,y ) x 2 2 x ≥− 1, and also that x 2 2 x →∞ if | x |→∞ . For any sequence ( x j ,y j ) with bardbl ( x j ,y j ) bardbl→∞ , either | x |→∞ (so f ( x,y ) x 2 2 x →∞ ) or | y |→∞ (so f ( x,y ) y 2 1 →∞ ); in either case, f ( x j ,y j ) →∞ . Since there exist such sequences with x j y j , the function is not bounded above. Now, if −→ s i = ( x i ,y i ) is a sequence with x i y i and f ( −→ s i ) inf x y f ( x,y ), either −→ s i have no convergent subsequence, and hence bardbl −→ s i bardbl→∞ , or some accumulation point of −→ s i is a local minimum for f . The first case is impossible, since we already know that then f ( −→ s i ) →∞ , while in the second case this accumulation point must be ( 1 2 , 1 2 ) , and then f ( −→ s i
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern