Questions tagged [extreme-value-theorem]

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69 questions
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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 $f_x = 0$: $0 = -2x$ $x=0$ ...
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$$f(x,y,z)=4x+2y+z$$ $$E=\{(x,y,z) \in R : (x+1)^2+4y^2+4z^2=4\}$$ 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 $g(x,... 1answer 57 views Local extremes of:$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}... 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 \... 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 ... 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 ... 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 ... 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]. ... 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 ... 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; ... 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 = 0x^2 + y^2 +z^2 -2x -2z +2 =0$$Solving functions with just 2 variables, ... 1answer 49 views Finding the maximum and minimum of f(x,y)= y(1-x^2-y^2) on D:={(x,y)|x^2+y^2 \leq 1} with extreme value theorem Define f : \Bbb R^2 \to \Bbb R by f(x,y)= y(1-x^2-y^2) Let D:={(x,y)|x^2+y^2 \leq 1}. Does f take a maximum and a minimum on D? If so in which points? So I calculated the critical points,... 1answer 143 views Applying Extreme Value Theorem to prove existence of unique fixed point Let K be a non-empty compact subset of \mathbb{R}^n, and let f:K \to K be a function which satisfies that \|f(x) - f(y)\|<\|x - y \| for all x,y\in K where x \neq y. I want to prove ... 0answers 33 views Rate of convergence of the maxium of a random sequence I came across a problem which requires the rate of convergence of \sup_{1\le j \le N} |\sum_{i=1}^{N}X_{i,j}/N|. If sup over a finite number of objects, this is a simple application of LLN. However, ... 2answers 371 views What is the critical points of f(x,y) = e^{\sin x\cos y} ? I try to find local extreme values and saddle point(s) of the f(x,y) = e^{\sin x\cos y} . But, when I take the partial derivative, I can't figure out to find critical points.$$f(x,y) = e^{\sin x\... 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, ... 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 ...