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Mathematik-Online problems:

Problem 116: Statements about Differentiable Functions

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Let $ I=(a,b)$ be a (nonempty) open interval and $ \xi$ a point within $ I$. The function $ f: I\longrightarrow\mathbb{R}$ shall be differentiable for $ I$, the function $ g: I\longrightarrow\mathbb{R}$ for $ I\setminus\{\xi\}$.

Mark which statements are true respectively false, and give reasons for your answers.

$ f^{(n+1)}(x)=0,\, \forall\, x\in I \ \Longrightarrow \ f$ is a polynomial of degree $ \leq n$  true $ \bigcirc $  false $ \bigcirc $
There exists at least one $ x\in I$ with $ f(x)=0$ or $ f'(x)=0$  true $ \bigcirc $  false $ \bigcirc $
$ f$ has a relative extremum for $ \xi$ $ \Longleftrightarrow$ $ f'(\xi)=0$  true $ \bigcirc $  false $ \bigcirc $
$ f(x)\neq 0,\, \forall\, x\in I$ $ \Longrightarrow$ $ 1/f^2$ has the derivation $ -2f'/f^3$  true $ \bigcirc $  false $ \bigcirc $
$ g$ is continuous at $ \xi$ $ \Longrightarrow$ $ fg$ is differentiable at $ \xi$  true $ \bigcirc $  false $ \bigcirc $
$ {\displaystyle{\lim_{x\to \xi+0}\, g'(x) = \lim_{x\to \xi-0}\, g'(x)}}$ $ \Longrightarrow$ $ g$ is differentiable at $ \xi$  true $ \bigcirc $  false $ \bigcirc $

(Authors: Apprich/Höfert)

see also:

[Solutions]
  automatisch erstellt am 14. 10. 2004