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\flushleft\underline{Homework 6 --- MATH 4590 Spring 2018} \\
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\item Chapter 3, \# 6 from the text: If $f \colon (a,b) \rightarrow \mathbb{R}$ assumes a maximum or minimum at some $\theta \in (a,b)$, prove that $f'(\theta)=0$. 

\item Prove using induction and the rules of differentiation (or directly, if you want) that the derivative of $x^n$ is $n x^{n-1}$. 

\item The notion of an ``absolute derivative" in metric spaces was recently defined: let $(M,d_1)$ and $(N,d_2)$ be metric spaces and let $f \colon M \rightarrow N$ be a function. The absolute derivative of $f$, $f^{|'|} \colon M \rightarrow [0,\infty)$, is defined by 
$$f^{|'|}(x) = \displaystyle\lim_{t \rightarrow x} \dfrac{d_2(f(x),f(t))}{d_1(x,t)}.$$
Fix a point $y \in N$ and let $f$ be the constant function $f(x)=y$. Prove that $f^{|'|}(x)=0$. 

\item Prove that if $f \colon \mathbb{R} \rightarrow \mathbb{R}$ (taking the usual metric in each case) is differentiable, then $f^{|'|}(x)=|f'(x)|$. $\Big($\textit{hint: you may find the fact that, for a function $g$, $\displaystyle\lim_{t \rightarrow x} |g(x)| = \left| \displaystyle\lim_{t \rightarrow x} g(x) \right|$ useful}$\Big)$
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