Math 6420 • PDE

Let
\[
f(x)=\operatorname{signum}(x)=
\begin{cases}
1, & x>0,\\
0, & x=0,\\
-1, & x<0.
\end{cases}
\]
\begin{enumerate}
\item[(a)] Find its first and second derivatives using the theory of distributions.

\item[(b)] Compute
\[
\begin{aligned}
I_0&=\int_{-1}^{\infty}\log(x+10)f(x)\,dx,\\
I_1&=\int_{-1}^{\infty}\log(x+10)f'(x)\,dx,\\
I_2&=\int_{-1}^{\infty}\log(x+10)f''(x)\,dx.
\end{aligned}
\]
\end{enumerate}