Math 6420 • PDE

Let \(f(x)=|x|-1\) on the interval \([-1,1]\).
\begin{enumerate}
\item[(a)] Treating \(f\) as a distribution, calculate
\[
  \int_{-1}^{1} f'(x)f'(x)\,dx
  \quad\text{and}\quad
  \int_{-1}^{1} f(x)f''(x)\,dx.
\]

\item[(b)] Explain why
\[
  \int_{-1}^{1} f'(x)f''(x)\,dx
  \quad\text{and}\quad
  \int_{-1}^{1} f(x)f'''(x)\,dx
\]
do not make sense.

\item[(c)] In general, what property is required of \(g(x)\) so that
\[
  \int_{-1}^{1} g(x)f^{(n)}(x)\,dx
\]
is well-defined for a given \(n\)th derivative?
\end{enumerate}