If a function is continuous on a closed interval, must it have an absolute maximum on that interval.?

Answer 1

True. a cts function takes compact spaces and connected spaces to compact and connected spaces (respectively).
Let's call A the closed interval.
A closed interval is both compact and connected. Therefore A's image under f is also compact and connected (thus a closed interval itself). Since every closed interval has a maximum, f(A) has a maximum (call it b). Since this maximum is the image of f, there must be some point (at least one) in A (call it a) such that f(a)=b.

Answer 2

Closed intervals are \textbf{compact} and the Heine-Borel theorem tells us that if f:D->R is continuous with D compact, then f(D) is compact.

Compact sets are closed and bounded so contain all their accumulation points. Hence, a function that is continuous on a closed and bounded interval attains its minimum and maximum values.

In differential calculus, we further learn that a critical number, c, is one for which f'(c)=0 or for which the derivative of f(x) at x=c does not exist. An amazing theorem tells us that the minimum and maximum values of a continuous function on a closed interval either occur at the endpoints of the interval or at the critical numbers on the interior of the interval.

So, you should keep taking math and take calculus and study it seriously, too! :)

Answer 3

real. If it does not have an absolute maximum it both has a limiteless cost or a reduce. If it has a reduce maximum (a cost V such that there is an x in the period such that V - f(x) < epsilon for any epsilon > 0) then through the indisputable fact that is closed there is an X with f(X) = V. If it has a limiteless cost (actual, if it has arbitrarily large values is a extra acceptable thanks to assert it) then you definately can opt for a chain of factors xi the position xi > 2^i. The reduce element of the xi is a member of the period through the indisputable fact that is closed, and the function isn't continuous at that element. for this reason it has a finite maximum cost through the indisputable fact that is continuous, and the point that maps to that cost is in the period through the indisputable fact that is closed. qed.

Answer 4

A horizontal line, for example, doesn't really have an absolute maximum. Also, if the graph has an asymptote, then it doesn't really have an absolute maximum. Then again, I'm not too sure.

Answer 5

yes. Even if it is monotonically increasing or decreasing, one of the end-points would be the maximum for the interval.

Answer 6

Yes. That's the "Extreme Value Theorem."

Answer 7

na
if its round over big area would be ok need other stuff too but