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Homework Help: A few problems and continuity and differentiability

  1. Oct 23, 2011 #1
    1. The problem statement, all variables and given/known data
    I have an upcoming math test, and these are from the sample exam. I'll post my solutions as I go along. I've submitted this post as is and am going to edit in my attempts. A few of these are "verify my proof is rigorous" others are "i've no idea what i'm doing

    1 Using simple algebra, prove that [itex]{\(x+y)^{\frac{1}{3}}<x^{\frac{1}{3}}+y^{\frac{1}{3}}}[/itex]for [itex]x>0,\ y>0[/itex]. Then prove that [itex[f(x)= x^{\frac{1}{3}}}[/itex] is uniforml\ y continuous on ${\ds (0,\infty)}$.
    2. Let $W$ be an open set in \mathbb{R}^n$. Let $\ds p\in W$ and $q \notin W$. Prove that there is a boundary point of $W$ on the line segment joining $p$ and $q$.
    3. Suppose that $\ds f'(x)$ exists on $(a,b)$ and $\ds f'(x) \ne 0$ on $(a,b)$. Prove that either $f'(x)>0$ for all $x\in (a,b)$ or $f'(x)<0$ for all $x \in (a,b)$. You may not assume that $f'$ is continuous. Hint: prove that $f$ is one-to-one.
    Let $f(x,y)$ denote the function defined by the following rules:
    $f(x,y)=\begin{cases} 0, \text{ if } (x,y)=(0,0)\\ \frac{xy}{\sqrt{x^2+y^2}}, \text{ if } (x,y)\ne (0,0)$

    Prove that $f$ is continuous at all points and has partial derivatives at
    all points, but is not differentiable at $(0,0)$.

    2. Relevant equations

    3. The attempt at a solution
    Last edited: Oct 23, 2011
  2. jcsd
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