{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Engineering Calculus Notes 331

Engineering Calculus Notes 331 - 3.6 EXTREMA 319 Theorem...

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

View Full Document Right Arrow Icon
3.6. EXTREMA 319 Theorem 3.6.6 (Extreme Value Theorem) . If S R 3 is compact, then every real-valued function f which is continuous on S achieves its minimum and maximum on S . Note that this result includes the Extreme Value Theorem for functions of one variable, since closed intervals are compact, but even in the single variable setting, it applies to functions continuous on sets more general than intervals. Proof. The strategy of this proof is: first, we show that f must be bounded on S , and second, we prove that there exists a point s S where f ( s ) = sup x S f ( x ) ( resp . f ( s ) = inf x S f ( x )). 12 Step 1: f ( x ) is bounded on S : Suppose f ( x ) is not bounded on S : this means that there exist points in S at which | f ( x ) | is arbitrarily high: thus we can pick a sequence s k S with | f ( s k ) | >k . Since S is (sequentially) compact, we can find a subsequence—which without loss of generality can be assumed to be the whole sequence—which converges to a point of S : s k s 0 S . Since f ( x ) is continuous on S , we must have f ( s k )
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern