extreme value theorem Definition and 14 Threads

For this problem, Why cannot we say that ##f(2.999999999) ≥ f(x)## and therefore absolute max at f(2.99999999999999) (without reasoning from the extreme value theorem)? Many thanks!

If ##f## is a constant function, then choose any point ##x_0##. For any ##x\in K##, ##f(x_0)\geq f(x)## and there is a point ##x_0\in K## s.t. ##f(x_0)=\sup f(K)=\sup\{f(x_0)\}=f(x_0)##. Now assume that ##f## is not a constant function. Construct a sequence of points ##x_n\in K## as follows...

Homework Statement Why does ##\lim_{n \rightarrow \infty} f(x_n) = f(c)## contradict ##\lim_{n \rightarrow \infty} \vert f(x_n) \vert = +\infty##? edit: where ##c## is in ##[a,b]## Homework Equations Here's the proof I'm reading from Ross page 133. 18.1 Theorem Let ##f## be a continuous real...

My textbook says the extreme value theorem is true for constants but I don't buy it. I mean I suppose that every value over a closed interval for a constant would be a maximum and a minimum technically but it seems like BS to me. Can anyone explain why this BS is true?

Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For 1a, I just took \lim_{{h}\to{0}} of the function using [ f(x+h)-f(x) / h ] and simplified. Ultimately, this gave me...

I was reading Wiki, I met a problem in understanding the the proof of boundedness theorem exactly when they said "Because [a,b] is bounded, the Bolzano–Weierstrass theorem implies that there exists a convergent subsequence" but Bolzano theorem state that if the sequence is bounded, which is...

Homework Statement Given g(x) = 1/x-2 is continuous and defined on [3, infinity). g(x) has no minimum value on the interval [3, infinity ). Does this function contradict the EVT? Explain. The Attempt at a Solution was wondering what would be a better explanation for this...

Homework Statement Show that the statement of the Extreme Value theorem does not hold if [a, b] is replaced by [a, b). Homework Equations The Attempt at a Solution Please help

Homework Statement This problem had been previously posted, however i have specific questions about it and don't feel that it was completely answered for it did not explain how you use the extreme value theorem in the problem and specifically, HOW DO YOU GET THE INTERVAL [a,b], The answer has...

Homework Statement Fix a positive number P. Let R denote the set of all rectangles with perimeter P. Prove that there is a member of R that has maximum area. What are the dimensions of the rectangle of maximum area? HINT: Express the area of an arbitrary element of R as a function of the...

Suppose f:[0,1]->R is continuous, f(0)>0, f(1)=0. Prove that there is a X0 in (0,1] such that f(Xo)=0 & f(X) >0 for 0<=X<Xo (there is a smallest point in the interval [0,1] which f attains 0) Since f is continuous, then there exist a sequence Xn converges to X0, and f(Xn) converges to f(Xo)...

From what i know if a graph has say one turning point, the relative global max(/min) is that point depending on the concavity correct? However as i was going through some notes, i notice that according to the mean value theorem that in a closed and bounded interval there exist a relative global...

Homework Statement If f is a continuous function on the closed interval [a,b], which of the following statements are NOT necessarily true? I. f has a minimum on [a,b]. II. f has a maximum on [a,b]. III. f'(c) = 0 for some number c, a < c < b Homework Equations Extreme Value...

Homework Statement Essentially, prove the Extreme Value Theorem. Homework Equations n/a The Attempt at a Solution Proof: Let a function f(x) be continuous on the closed interval [a,b]. Moreover, define a set A such that A={x ϵ [a,b]}. Since f(x) satisfies the condition for the...