Let \(Q=\{(x_1,x_2):x_j>0\}\) (the first quadrant in \(\mathbb R^2\)) and let \(V=\{(y_1,y_2):y_1>0\}\) (the right half plane – RHP). Show \(f\) maps \(Q\) bijectively onto \(V\) and determine the inverse (of \(f|_Q:Q\to V\)).

Show if \(\det(Df(c))=0,\) then \(f\) is not one-one on any neighborhood of \(c.\) [This conclusion does not follow from the Inverse Function Theorem.]

Consider the function $f:\mathbb R^2 \to \mathbb R^2$ defined by $f(x,y)= (xy,x-y).$ Compute $Df(x,y)$. Verify that the inverse function theorem applies at the point $(1,1)$. What is the conclusion?

Let $Q=\{(x_1,x_2):x_j>0\}$ (the first quadrant in $\mathbb R^2$) and let $V=\{(y_1,y_2):y_1>0\}$ (the right half plane – RHP). Show $f$ maps $Q$ bijectively onto $V$
and determine the inverse (of $f|_Q:Q\to V$).

Show if $\det(Df(c))=0,$ then $f$ is not one-one on any neighborhood of $c.$ [Note: This conclusion does not follow from the Inverse Function Theorem.]

Suppose $U\subset\mathbb R^n$ is open, $c\in U$ and
$f:U\to\mathbb R$. Define {\it $f$ has a local minimum at $c$}
and prove that if $f$ has a local minimum at $c$ and $f$
is differentiable at $c$, then $Df(c)=0$. [Suggestion : show $Df(c)u=0$ for all unit vectors $u.$]

• Show, directly from the definition, that \( f:\mathbb R^2\to \mathbb R^2\) defined by \( f(x_1,x_2) = (x_1^2-x_2^2, 2x_1x_2)\) is differentiable and compute its derivative (at each point). At which points \(c\) does the derivative \(Df(c)\) fail to be invertible?
  Show, directly from the definition, that $f:\mathbb R^2\to \mathbb R^2$ defined by $f(x_1,x_2) = (x_1^2-x_2^2, 2x_1x_2)$ is differentiable and compute its derivative (at each point). At which points $c$ does the derivative $Df(c)$ fail to be invertible?

• Suppose \(A\) is an \( m\times n\) matrix and \(a\) is a vector in \(\mathbb R^m\). Show that \( f:\mathbb R^n\to \mathbb R^m\) defined by \( f(x) = Ax + a\) is differentiable and \(Df(c)=A\) (for all \(c\)).
  Suppose $A$ is an $m\times n$ matrix and $a$ is a vector in $\mathbb R^m$. Show that $f:\mathbb R^n\to \mathbb R^m$ defined by $f(x) = Ax + a$ is differentiable and $Df(c)=A$ (for all $c$).

Show \( \||S\|=c.\) Show also that \(I-S\) is invertible (even if \(c\ge 1\)) and compare with Lemma 14.17. (For inspiration you might look at Problem 14.7.)

Fix $c>0$ and let
\[
S= c \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{pmatrix}.
\]

Show $\|S\|=c.$ Show also that $I-S$ is invertible (even if $c\ge 1$)
and compare with Lemma~\ref{lem:Iminusnorm}. (For inspiration you might look at Problem 14.7.)

Consider, for $s\in\mathbb R$, the series $\sum_{n=1}^\infty (-1)^n n^{-s}.$
Show this series converges absolutely for $s\in (1,\infty),$ conditionally for $s\in (0,1]$
and diverges for $s\in (-\infty,0].$ Finally, show, given $1<S$,
that the sequence of partial sums
$d_n(s)=\sum_{j=1}^n (-1)^j j^{-s}$ converges uniformly on $[S,\infty).$

Suppose $f:[0,\infty)\to [0,\infty)$ is continuous and decreasing. Show, $\sum_{j=0}^\infty f(j)$ converges if and only if $F(x)=\int_0^x f \, dt$ is bounded above independent of $x$ (meaning there is an $M$ such that $F(x) \le M$ for all $x\ge 0$).

Suppose $(a_n)$ a decreasing sequence of positive numbers. Use Problem 12.1 to show the series $\sum_{j=0}^\infty a_j$ converges if and only if the series $\sum_{j=0}^\infty 2^j a_{2^j}$ converges. (You might find Exercise 12.1 useful.)

Suppose $f:[a,b]\to \mathbb R$ is continuous and, for $n\in\mathbb N^+\cup\{\infty\},$ let $\|\cdot\|_n$ denote the $L^n([a,b])$ norm. Prove that
\begin{equation*}
\lim_{n\to\infty} \|f\|_n =\|f\|_\infty.
\end{equation*}

Suppose $f:[-1,1]\to\mathbb R$ takes nonnegative values. Show, if $f\in \mathcal{RI}([-1,1]),$ is continuous at $0$ and if $f(0)>0$, then
\[
\int_{-1}^1 f \, dx >0.
\]
Suggestion: First show First show there is a $\delta>0$ such that $f(x)\ge \frac{f(0)}{2}$ for $|x|\le \delta.$ Then show there is a partition $P$ such that $L(P,f)>0$ and conclude the lower integral is positive.

Suppose $f:\mathbb R\to \mathbb R$ is differentiable. Show, if $|f^\prime(t)|<1$ for all $t$, then $f$ has at most one fixed point (a point $y$ such that $f(y)=y$). Show, if there is an $0\le A<1$ such that $|f^\prime(t)|\le A$ for all $t$, then $f$ has exactly one fixed point. [As a suggestion for the second part, choose any point $x_1,$ let $x_{n+1}=f(x_n)$ and use Proposition 5.12.]

Suppose $f:[0,1]\to\mathbb R$ is continuous and let $g_n:[0,1]\to\mathbb R$ denote the function $g_n(t) =t^nf(t)$. Show, if $(g_n)$ converges uniformly, then $f(1)=0$; and conversely, if $f(1)=0$, then $(g_n)$ converges uniformly.

Suppose $X$ is a set and $(f_n)$ is a sequence of functions $f_n:X\to\mathbb R.$ Show if each $f_n$ is bounded and $(f_n)$ converges uniformly to a function $f:X\to \mathbb R,$ then $(f_n)$ is uniformly bounded; that is, there is an $M$ such that $|f_n(x)|\le M$ for all $x\in X$ and $n.$ (Note: it also follows that $|f(x)|\le M$ for all $x\in X.$)

Homework grading abbreviations