Questions tagged [extreme-value-theorem]

This tag is for questions relating to "Extreme Value Theorem". The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions.

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9
votes
2answers
3k views

$C^1$ function on compact set is Lipschitz

Let $\emptyset\ne A\subset\mathbb{R}^n$ be open and let $f \in C^1 (A, \mathbb{R})$ be a function. Let $\emptyset\ne K\subset A$ be compact and convex. I want to prove that $f$ is Lipschitz on $K$; ...
8
votes
2answers
274 views

Trouble understanding how the Transfer Principle is applied for the Extreme Value theorem.

I am reading Keisler's Elementary Calculus (which can be downloaded here). I am having trouble understanding his proof sketch of Extreme Value Theorem and how he is applying the Transfer Principle. ...
4
votes
5answers
2k views

Finding extreme values where second derivative is zero

Consider this function: $$f(x)= 5x^6 - 18x^5 + 15x^4 - 10$$ I am told to find the extreme values of this function. So at first, I take the first derivative and set it zero. $$f'(x)=30x^5-90x^4+60x^...
3
votes
2answers
64 views

Open sets in a particular norm.

Let $f \in C^2[0,1]$, define $||f||_{2,\infty} := \underset{x \in [0,1]} {\text{sup}} |f(x)|+ \underset{x \in [0,1]} {\text{sup}}|f'(x)|+ \underset{x \in [0,1]} {\text{sup}}|f''(x)|$. Then I wish to ...
3
votes
2answers
47 views

Minimum distance from the points of the function $\frac{1}{4xy}$ to the point $(0, 0, 0)$

I am trying to find the minimum distance from the points of the function $\large{\frac{1}{4xy}}$ to the point $(0, 0, 0)$. This appears to be a problem of Lagrange in which my condition: $C(x,y,z) = ...
3
votes
0answers
88 views

Extreme value theory - proof this is a poisson point process

Let $(X_n)_{n \geq 1}$ be an i.i.d sequence of real valued RVs with continous distribution function f and $M_n:=\max \{X_1,...,X_n \}$. Let $U_n:=\inf \{ k \in \mathbb{N} | X_k>X_{U_{n-1}} \}$ be ...
2
votes
2answers
511 views

Prove that a function attains its minimum

Let $f$ be a real continuous function defined on $D=[0,+\infty)$, $f(x)\geq 0$ for all $x\in D$, and $\lim_{x\to+\infty}\:f(x)=+\infty$. Prove that $f(x)$ attains its minimum on $D$. Idea for a ...
2
votes
3answers
50 views

Show $f(x,y) = y^2 - x^2$ at $(0,0)$ has a critical point, but is not a max/min value

So as always... I found the partial derivative with respect to $x$ and $y$ of $f(x,y)$ which gave me: $f_x=-2x$ $f_y=2y$ So I wasn't too sure what to do next, but I set $f_x = 0$: $0 = -2x$ $x=0$ ...
2
votes
2answers
561 views

If a function is defined on a closed interval $[a,b]$, does it necessarily achieve a max and min value on that interval?

The extreme value theorem requires that a function be continuous on a closed interval $[a,b]$ for it to necessarily take on a max and min, but I've been thinking and it seems to me that as long as it ...
2
votes
1answer
162 views

Applying Extreme Value Theorem to prove existence of unique fixed point

Let $K$ be a non-empty compact subset of $\mathbb{R}^n$, and let $f:K \to K$ be a function which satisfies that $\|f(x) - f(y)\|<\|x - y \|$ for all $x,y\in K$ where $x \neq y$. I want to prove ...
2
votes
1answer
32 views

Question about a certain part of Weierstrass extreme value theorem

The lemma says that if $A \in \mathbb{R}^{n}$ is compact and $f:A \to \mathbb{R}$ is continuous in $A$, then there's a real $M$ such that $\forall X \in A,f(X) \leq M$. My textbook's proof goes this ...
2
votes
1answer
38 views

Absolute conditional minimum of function in n-dimensional space

Function $$F(x_1,x_2,...,x_n) = \sum_{i=1}^n x_i$$ on the constraint $$G(x_1,x_2,...,x_n)=\prod_{i=1}^n x_i-1$$ what I come up with is writing down the Lagrangian: $$L = F(x_1,x_2,...,x_n) - \...
2
votes
1answer
1k views

Proving that any continuous complex function is bounded on a closed bounded set

Let $E$ be a closed, bounded set and let $f(z)$ be a continuous complex function in $E$. Prove that $f(z)$ is bounded in $E$. I began the argument the same way that the boundness theorem is addressed ...
2
votes
0answers
17 views

Tail of CDF of non-central chi squared RV using asymptotics of Bessel function

The pdf and cdf of the non-central chi squared RV (under the scenario I am studying) is given as follows: \begin{align} &f(x)=\frac{1}{v} \exp\left(\frac{-(a+x)}{v}\right)I_{0}\left(\frac{\sqrt{xa}...
2
votes
2answers
70 views

Can the ratio of the two smallest element of an iid sample converge to 1 if the support of $X$ is positive?

We have: $\mathbb P(X \leq 0)=0$ and $\mathbb P (X \leq a)>0$ for any $a>0$.
1
vote
2answers
67 views

How do we construct a continuous function on the interval $(0, 1]$ without a minimum or a maximum?

One of my Calculus lecture videos poses the following challenge (shortly after an exposition of the Extreme Value Theorem): Construct a function $f$ such that $f$ is continuous on $(0, 1]$ ...
1
vote
1answer
262 views

Let $f(x)$ be a continuous function on $[0,1]$ and $f(0) = f(1)$. Let $α ∈ (0,1)$. Prove that there exists an $x ∈ (0,1)$ such that $f(x) = f(αx)$.

Suppose $f(x)$ is a continuous function on $[0,1]$ with $f(0) = f(1)$. Let $α ∈ (0,1)$. Prove that there exists an $x ∈ (0,1)$ such that $f(x) = f(αx)$. I tried $h(x) = f(x) - f(αx)$ and Intermediate ...
1
vote
2answers
38 views

Find the minimum value of the function a(x).

$$a(x)= \sqrt{x^3}+\sqrt{x^{-3}}-4(x+\frac{1}{x})$$ One of the ways I could think of was to find out the global extreme values and proceed.But as I began doing it that it takes a lot of ...
1
vote
2answers
140 views

Show that if f is continuous and periodic then f attains both its minimum and its maximum.

Show that if f is continuous and periodic then f attains both its minimum and its maximum. The solution is given below: But I wonder why he choose the k like this and is the solution does not ...
1
vote
4answers
81 views

Difficulty finding Lagrange multiplier because of $\leq$

Let $f: \mathbb R^3 \to \mathbb R$ be defined by $$f(x,y,z)=x-y+z$$ and $$E:=\{(x,y,z)\in \mathbb R^{3} \mid x^2+2y^2+2z^2\leq1\}$$ Find the extrema of $f$ on $E$. Path: I have already proven that ...
1
vote
2answers
552 views

Intermediate value theorem on infinite interval $\mathbb{R}$

I have a continious function $f$ that is strictly increasing. And a continious function $g$ that is strictly decreasing. How to I rigorously prove that $f(x)=g(x)$ has a unique solution? Intuitively, ...
1
vote
3answers
269 views

Prove that a positive monic polynomial with even degree has a minima but not a maxima.

Prove that a positive monic polynomial with even degree has a minima but not a maxima. You may use the fact that there are finitely many critical points. Let's take the worst case where the first ...
1
vote
2answers
83 views

Show the existence of a global maximum of a continuous function with unbounded domain

I am given a function $f(t) \in \mathbb{R}$ which is continuous; bounded above by $M$ and below by $0$. $f$ is differentiable everywhere except at $f=0$. Also, $\lim_{t \to \infty} f = 0$ and $t \in [...
1
vote
1answer
109 views

What does it mean centering a Gumbel distribution?

Consider $M$ i.i.d. random variables $V_1,..., V_M$ distributed as Gumbel with location $\lambda$ and scale $\beta$. We know that (see proof at the end of the question) $$ E(\max_{k\in \mathcal{Y}} ...
1
vote
2answers
212 views

Extreme values for a vector equation

For a question on physics.stackexchange about Does the Ampère-Maxwell law fail for the field of a uniformly moving point charge? with $$ \vec B(P) = \dfrac{\mu_0 q}{4 \pi} \dfrac{1 - v^2/c^2}{[1 - (v^...
1
vote
1answer
59 views

Local extremes of: $f(x,y) = (x^2 + 3y^2)e^{-x^2-y^2}$

I am looking to find the local extremes of the following function: $$f(x,y) = (x^2 + 3y^2)e^{-x^2-y^2}$$ What have I tried so far? Calculate the partial derivatives: $$\frac{\partial f}{\partial x}...
1
vote
2answers
65 views

if $A,B \subseteq \Bbb R^n, A \cap B = \emptyset$ , $A$ compact and $B$ closed then the distance is achieved.

For 2 sets $A,B \subseteq \Bbb R^n$ such that $A \cap B$ = $\emptyset$, denote: $d(A,B):=$ inf$\{||x-y|| : x\in A, y \in B \}$. Show that if $A$ is compact and $B$ is closed, then there exists $a^*...
1
vote
2answers
455 views

What is the critical points of $f(x,y) = e^{\sin x\cos y} $?

I try to find local extreme values and saddle point(s) of the $f(x,y) = e^{\sin x\cos y} $. But, when I take the partial derivative, I can't figure out to find critical points. $$f(x,y) = e^{\sin x\...
1
vote
2answers
31 views

For what values of parameter m the function $g(x) - 2x^3 - 3x^2 + mx + 3 $ has an extremum of 10? An easier way to solve it

My attempt to solve this problem is very tedious and I do not think it is the optimal method. $$g'(x) = 6x^2 -6x + m$$ $$g'(x) = 0 \Rightarrow x = \frac{3 \pm\sqrt{9-6m} }{6}$$Now, I need to subsitute ...
1
vote
1answer
13 views

Stationary points problem

I already made a first derivation of $f\left ( s,t \right )$. For $\frac{\partial f}{\partial s}=4s^{3}-2s-2t$ and for $\frac{\partial f}{\partial t}=4t^{3}-2s-2t$. I have to find the stationary ...
1
vote
1answer
85 views

Show that a continuous periodic function on $\mathbb{R}$ attains its maximum and minimum. - Proof Verification

Let $f$ be a continuous periodic function on $\mathbb{R}$, that is $\exists\ d > 0$ s.t $f(x+d) = f(x)\ \forall x \in \mathbb{R}$. I believe I have the whole idea of how to prove this, but I'm ...
1
vote
1answer
54 views

How to find extreme values of an $f(x,y)$ function?

I need this for my semester exams, unfortunately I was absent the day this topic was "talked about". My function is the real-valued $$f(x,y)=x-xy+x^2+y^2$$, interpreted on $\mathbb{R}^2$. Single-...
1
vote
1answer
53 views

Second derivative test in the Hilbert space case

Let $H,E$ be $\mathbb R$-Hilbert spaces; $f\in C^2(\Omega)$; $c\in C^2(\Omega,E)$; $M:=\left\{c=0\right\}$; $x\in M$ be a local minimum of $f$ constrained on $M$, i.e. $$f(x)\le f(y)\;\;\;\text{for ...
1
vote
2answers
34 views

Example of a calculus optimization problem where the answer occurs at an endpoint

I'm teaching optimization problems in calculus right now. An easy example would be something like: Find the dimensions of a rectangle with perimeter $100$ m whose area is as large as possible. The ...
1
vote
1answer
22 views

Proof Verification for Extreme Value Theorem

I had an idea for a proof of the Extreme Value Theorem, and I was wondering if it was valid. Any advice you have would be greatly appreciated. Thank you! Prove: If a function f(x) is continuous on a ...
1
vote
0answers
56 views

Domain of attraction $F(x)=\exp(-x-\sin(x))$

I need to show that $F(x)=\exp(-x-\sin(x)),x >0$ is in no domain of attraction. That means that there exist no $a_{k} >0, b_{k} \in \mathbb{R}, k \in \mathbb{N}$ with $\lim\limits_{k \to \infty}...
1
vote
0answers
32 views

Critical point of a multivariable function

For the function $f(x,y)=x^2y^2 + \frac{1}{y} + \ln(\frac{1}{x})$ I get two critical points, namely $P_1 =\left(\sqrt{\frac{1}{2}},1 \right)$ and $P_2 =\left(-\sqrt{\frac{1}{2}},1 \right)$. However ...
1
vote
0answers
34 views

Rate of convergence of the maxium of a random sequence

I came across a problem which requires the rate of convergence of $\sup_{1\le j \le N} |\sum_{i=1}^{N}X_{i,j}/N|$. If sup over a finite number of objects, this is a simple application of LLN. However, ...
1
vote
0answers
89 views

Frechet distribution function compared to empirical data

So basically I am trying to evaluate VaR on the Tesla stock using a Block Maxima method. I.e. I assume that the worst weekly log returns follow a generalized extreme value distrubtion with the shape ...
0
votes
3answers
56 views

Suppose a continuous function attains its minimum, prove that the function is not injective

Suppose a continuous function $f: (0,2) \to \mathbb R $ attains its minimum at $x_0 \in (0,2)$, prove that the function is not injective. We need to show there are some $a$ and $b$ such that $f(a)=f(...
0
votes
2answers
51 views

Find out if a function has a maximum and minimum or not

$\psi: \mathbb{R}$ → $\mathbb{R}$ is a continuous function such that $\lim_{x\to +∞} ψ(x) = +\infty $ and $\lim_{x\to -∞} ψ(x) =-\infty $ Decide if G: $\mathbb{R}$ → $\mathbb{R}$, $G(x) = \frac{ψ(x)}{...
0
votes
3answers
49 views

Extreme values $f(x)=(x-2)^{\frac{1}{3}}$

find all points of intrest for the function: $f(x)=(x-2)^{\frac{1}{3}}$ Here we can clearly see that when $x=2$ $f(x)=0$ so I know that there atleast should exist a critical point. Since the ...
0
votes
3answers
586 views

Find extreme values of an implicit function

The prompt is to find the extreme values of an implicit function $z(x, y)$ The functions are $$ x^2 + y^2 + z^2 -3z = 0$$ $$x^2 + y^2 +z^2 -2x -2z +2 =0$$ Solving functions with just 2 variables, ...
0
votes
1answer
32 views

Local extreme of function : $f\left ( x,y \right) =x^{2}+2x+y^{2}-2y+3$ [closed]

I am supposed to find the local extreme of function : $f\left ( x,y \right) =x^{2}+2x+y^{2}-2y+3$ on circle with the origin in the coordinate axis and radius $2\sqrt{2}$. I know how should I do this, ...
0
votes
2answers
53 views

Extreme value theorem: help with contradiction

I have a problem understanding the last part of the usual proof of the extreme value theorem (found for example here: Extreme Value Theorem proof help) It is this part that I have trouble ...
0
votes
2answers
74 views

Generalized Way of Treating Extrema under Certain Constraints (Inequalities)

Let's take a simple example $f: \mathbb R^{2} \to \mathbb R$, $f(x,y)=xy$ and then I want to treat $f$ for a constraint $M$ under all possible inequalities: Case 1) $M:=\{(x,y)\in \mathbb R^{2}|x^2+y^...
0
votes
1answer
60 views

Absolute Extremes of: $f(x,y,z) = xyz$ with $x+y+z=1$

I am attempting to find the absolute extremes of the function: $$f(x,y,z) = xyz$$ with the condition that: $$x+y+z=1$$ So far I have gathered the following: Condition: $$C(x,y,z) = x+y+z-1$$ and ...
0
votes
2answers
53 views

Determine the Conditional Extremes of a Function

I am trying to determine the conditional extremes the following question: Determine the point of the plane, $2x-y+2z=16$ closest to the origin. but I do not fully understand the question. If I am ...
0
votes
1answer
40 views

I need to clarify my understanding of the extreme value theorem.

I have had some difficulty in trying to understand the extreme value theorem but I think I might understand it correctly now but would like to clarify that I am thinking about this correctly. If we ...
0
votes
1answer
64 views

Reference for extreme value theorem.

I look for a reference of the Extreme Value Theorem for semicontinuous functions defined on a topological space. I know the proof, but I want to cite this result in my work.