Existence of golobal minimum point for continuous functions on $\mathbb{R}^2$
Let
1. $g:[0,\infty)\rightarrow \mathbb{R}$ such that $\lim_{t\rightarrow \infty}g(t)=\infty$
2. $f: \mathbb{R}^2\rightarrow \mathbb{R}$ continues such that for all (x,y): f(x,y) $\geq$ g($\sqrt{x^2+y^2} $ ).
Prove that f has a global minimum point.
Answer
Answers can only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
1 Attachment
Mathe
3.6K
-
In line 7 I'm not sure how f(x min,y min) can be little/equal to c if for all (x,y) f(x,y)>=g((x^2+y^2)^0.5) and g>c?
-
Nevermind. I got it
-
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 474 views
- $10.00
Related Questions
- Partial Derivatives and Graphing Functions
- Convex subset
- Calculus 3 Challeng problems
- Let $ X = x i+ y j+z k$, and $r=||X||$. Prove that $\nabla (\frac{1}{r})=-\frac{X}{r^3}.$
- Find $n$ such that $\lim _{x \rightarrow \infty} \frac{1}{x} \ln (\frac{e^{x}+e^{2x}+\dots e^{nx}}{n})=9$
- Show that the line integral $ \oint_C y z d x + x z d y + x y d z$ is zero along any closed contour C .
- Finding Binormal vector from the derivative of the Normal and Tangent.
- Triple integral
Is g continuous?
We don't know. Any way its not a given
Got it! No assumption on g!