What is extreme value theorem: Definition and 14 Discussions
In calculus, the extreme value theorem states that if a real-valued function
f
{\displaystyle f}
is continuous on the closed interval
[
a
,
b
]
{\displaystyle [a,b]}
, then
f
{\displaystyle f}
must attain a maximum and a minimum, each at least once. That is, there exist numbers
c
{\displaystyle c}
and
d
{\displaystyle d}
in
[
a
,
b
]
{\displaystyle [a,b]}
such that:
The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function
f
{\displaystyle f}
on the closed interval
[
a
,
b
]
{\displaystyle [a,b]}
is bounded on that interval; that is, there exist real numbers
m
{\displaystyle m}
and
M
{\displaystyle M}
such that:
This does not say that
M
{\displaystyle M}
and
m
{\displaystyle m}
are necessarily the maximum and minimum values of
f
{\displaystyle f}
on the interval
[
a
,
b
]
,
{\displaystyle [a,b],}
which is what the extreme value theorem stipulates must also be the case.
The extreme value theorem is used to prove Rolle's theorem. In a formulation due to Karl Weierstrass, this theorem states that a continuous function from a non-empty compact space to a subset of the real numbers attains a maximum and a minimum.
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...