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.

View More On Wikipedia.org
Recent contents Top users
  1. C

    Why is continuity necessary before applying the Extreme Value Theorem?

    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!
  2. Eclair_de_XII

    B Is this a valid proof for the Extreme Value Theorem?

    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...
  3. F

    Extreme value theorem, proof question

    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...
  4. M

    Extreme Value Theorem true for constants?

    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?
  5. A

    MHB Limit Definitions and Extreme Value Theorem Help Needed

    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...
  6. A

    MHB Extreme value theorem Proof

    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...
  7. I

    Extreme value theorem question?

    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...
  8. M

    Extreme Value theorem does not hold if [a; b)

    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
  9. T

    Extreme Value Theorem: Maximum Area of Rectangle

    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...
  10. L

    Using the Extreme Value Theorem on rectangles?

    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...
  11. C

    Is there a smallest point in the interval [0,1] where f attains the value of 0?

    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)...
  12. S

    Need to clarify Extreme value theorem

    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...
  13. C

    Extreme Value Theorem & MVT/Rolles Theorem

    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...
  14. J

    Proof of the Extreme Value Theorem

    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...
Back
Top