\documentclass{exam}
\newcommand{\poly}{\textit{\textbf{P}}}

\begin{document}

\begin{center}
\textbf{\Large MATH 3912 - Assignment 5}
\end{center}
\normalsize

\begin{questions}

\vfill
\question
Suppose $p \in \poly_2$ and $p(x_0) = p'(x_0) = p''(x_0) = 0$. Show that $p$ must be identically zero.

\vfill
\question
Suppose $p \in \poly_2$ and suppose that $p(x_1) = p'(x_1) = 0$ and $p(x_2) = 0$ for some fixed numbers $x_1 \ne x_2$. Show that $p$ must then be identically zero.

\vfill
\question
Show that if $p \in \poly_n$ and $p$ has a root of total multiplicity $n+1$ then $p$ must be identically zero. Recall that $p$ has a root of multiplicity $n+1$ at $x = a$ if 
$p(a) = p'(a) = \dots = p^{(n)}(a) = 0$

\vfill
\question
If $p(x)$ is a polynomial of even degree, what is $lim_{x \rightarrow \infty} p(x)$ and 
$lim_{x \rightarrow -\infty} p(x)$? Evaluate the same limits in case $p$ is a polynomial of odd degree.

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\question
If $f(x)$ is a polynomial then $lim_{n \rightarrow \infty} f^{(n)}(x) = 0$ for all $x$.

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\question
Show that $f(x) = 2^x$ can coincide with a polynomial at only a finite number of points.

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\question
Suppose $f$ is real analytic for all $x$ and $f^{(k)}(x) > 0$ for $k=0, 1, \dots$. Then $f$ can not coincide with a polynomial infinitly often.

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\question
Suppose $X = C^1([a,b])$ and $x_0$ is a fixed point in $[a, b]$. Define $L(f) = f'(x_0)$ for all 
$f \in X$. Is $L$ a linear functional?

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\question
Suppose $X = C^0([a,b])$. Define two functionals $L$ and $G$ via
\begin{displaymath}
L(f) = \int_a^b x^2 f(x) dx \hbox{  and  } 
G(f) = \int_a^b \left(f(x)\right)^2 dx
\end{displaymath}
Which of these functions is linear?

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\question
Suppose $X = \poly^n$, $x_0$ is a fixed point in $[a, b]$, and $f$ is a continous function defined on the 
interval $[a, b]$. Define $L(p) = (f \circ p)(x_0)$. Is $L$ a linear functional?

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\question
Suppose $X = C^1([a,b])$. Define $L(f) = f'$ for all $f \in X$. What is the range of this map $L$? Is the map $L$ linear? Is $L$ a linear functional?

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\question
Let $X$ be the space of $n \times n$ matrices and define $L(A) = \det(A)$. Is $L$ a linear functional?

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\question
Let $X$ be the space of $n \times n$ matrices. For a matrix $A \in X$ define the trace of A as
\begin{displaymath}
trace(A) = trace
\left(
\begin{array}{cccc}
a_{1,1} & a_{1,2} & \dots & a_{1,n} \\
a_{2,1} & a_{2,2} & \dots & a_{2,n} \\
 \vdots & \vdots & \ddots & \vdots \\
a_{n,1} & a_{n,2} & \dots & a_{n,n} \\
\end{array}
\right) = 
a_{1,1} + a_{2, 2} + \dots + a_{n,n}
\end{displaymath}

Is $L(A) = trace(A)$ a linear operator?


\vfill
\end{questions}
\end{document}

