25. Let
$F(x) = \begin{cases} x - 2, & x > 0, \\ 0, & x < 0. \end{cases}$
Show that $F''(x) = 0$ for all $x \neq 0$, and $\int_{-\infty}^{\infty} F''(x) dx = 1$, which leads you to think
that $F''(x)$ might $= \delta(x)$. Show in two ways, as outlined in (a) and (b), that this is
not true.
(a) Show that $\int_{-\infty}^{\infty} \phi(x) F''(x) dx = \phi(0) + 2\phi'(0)$, where $\phi$ is any test function.
Then by (11.6) and (11.14), what is $F''(x)$?
(b) Show that $F(x) = (x - 2)u(x)$ where $u(x)$ is the unit step function in (11.17).
Differentiate this equation twice and simplify using (11.17) and (11.18). Compare your result in (a).
(c) As in (a) and (b), find $G''(x)$ in terms of $\delta$ and $\delta'$ if
$G(x) = \begin{cases} 3x + 1, & x > 0, \\ 2x - 4, & x < 0. \end{cases}$
