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Questions tagged [extreme-value-theorem]

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9
votes
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 $K$; ...
8
votes
2answers
235 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. ...
1
vote
4answers
74 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 ...