Questions tagged [extreme-value-theorem]

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2
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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 $f_x = 0$: $0 = -2x$ $x=0$ ...
0
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1answer
60 views

Find maximum and minimum value of function of three variables on the set $E$

$$f(x,y,z)=4x+2y+z$$ $$ E=\{(x,y,z) \in R : (x+1)^2+4y^2+4z^2=4\}$$ I know I should write here what I already did but I could come up with literally nothing. Should I just find extreme values of $g(x,...
1
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1answer
57 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
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2answers
58 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 \...
2
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1answer
840 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 ...
0
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1answer
52 views

Show that f has a minimun

been trying to solve this for some time now. f is continuous in $ [0,\infty), $ and $\lim_{x\to \infty}f(x) = L . $ prove that if there exist $x \ge 0 $ such that f(x) < L then f has a minimum ...
0
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1answer
54 views

Several questions about continuous, derivative and extrema

Those problems come with my proof of question. I already found a better solution for this question, but there exists some confusion in the first proof occur to my head Original Question f(x) is ...
0
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1answer
37 views

Extremum of a sum of polynomial and square root of polynomial

Let $f(x)$ be of the form $f(x) = P_1(x)+\sqrt{P_2(x)}$, where $P_1(x)$ is a monomial $P_1(x)=ax+b$ and $P_2(x)$ is a quadratic function $P_2(x)=cx^2+dx+d$, defined on the closed interval $[0,1]$. ...
0
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2answers
101 views

Extreme value theorem for $f:\mathbb{R}^n\to \mathbb{R}^m$

A marker comments that the EVT only considers functions of the form $f:\mathbb{R}^n\to \mathbb{R}$. However, I don't understand why this should be the case. For there is, for example, the notion of a ...
9
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2answers
2k 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 $K$; ...
0
votes
3answers
508 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
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1answer
49 views

Finding the maximum and minimum of $f(x,y)= y(1-x^2-y^2)$ on $D:=${$(x,y)|x^2+y^2 \leq 1$} with extreme value theorem

Define $f : \Bbb R^2 \to \Bbb R$ by $f(x,y)= y(1-x^2-y^2)$ Let $D:=${$(x,y)|x^2+y^2 \leq 1$}. Does $f$ take a maximum and a minimum on $D$? If so in which points? So I calculated the critical points,...
2
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1answer
143 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 ...
1
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0answers
33 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
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2answers
371 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
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2answers
371 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
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2answers
29 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
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0answers
76 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 ...
1
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3answers
245 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 ...