Let $f_0(x) = x$. For $x > 0$, define functions inductively by $f_{n+1}(x) = x^{f_n(x)}$. So $f_1(x) = x^x$, $f_2(x) = x^{x^x}$, etc. Note that $f_0(1) = f_0'(1) = 1$.
Statements
(25) As $x \rightarrow 0^+$, $f_1(x) \rightarrow 1$. (26) As $x \rightarrow 0^+$, $f_2(x) \rightarrow 1$. (27) $\int_0^1 f_3(x)\, dx = 1$. (28) The derivative of $f_{123}$ at $x = 1$ is 1.

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is equal to
(A) $f _ { n } ( x )$
(B) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(C) $f _ { n } ( x ) f _ { n - 1 } ( x ) \cdots f _ { 1 } ( x )$
(D) $f _ { n + 1 } ( x ) f _ { n } ( x ) \cdots f _ { 1 } ( x ) e ^ { x }$

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is:
(a) $f _ { n } ( x )$
(b) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(c) $f _ { n } ( x ) f _ { n - 1 } ( x ) \ldots f _ { 1 } ( x )$
(d) $f _ { n + 1 } ( x ) f _ { n } ( x ) \ldots f _ { 1 } ( x ) e ^ { x }$.

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is:
(a) $f _ { n } ( x )$
(b) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(c) $f _ { n } ( x ) f _ { n - 1 } ( x ) \ldots f _ { 1 } ( x )$
(d) $f _ { n + 1 } ( x ) f _ { n } ( x ) \ldots f _ { 1 } ( x ) e ^ { x }$.

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is equal to
(A) $f _ { n } ( x )$
(B) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(C) $f _ { n } ( x ) f _ { n - 1 } ( x ) \cdots f _ { 1 } ( x )$
(D) $f _ { n + 1 } ( x ) f _ { n } ( x ) \cdots f _ { 1 } ( x ) e ^ { x }$

Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is equal to
(A) $f _ { n } ( x )$
(B) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(C) $f _ { n } ( x ) f _ { n - 1 } ( x ) \cdots f _ { 1 } ( x )$
(D) $f _ { n + 1 } ( x ) f _ { n } ( x ) \cdots f _ { 1 } ( x ) e ^ { x }$

Let $$\begin{gathered} p ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 2 x , x \in \mathbb { R } \\ f _ { 0 } ( x ) = \begin{cases} \int _ { 0 } ^ { x } p ( t ) d t , & x \geq 0 \\ - \int _ { x } ^ { 0 } p ( t ) d t , & x < 0 \end{cases} \\ f _ { 1 } ( x ) = e ^ { f _ { 0 } ( x ) } , \quad f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) } , \quad \ldots \quad , f _ { n } ( x ) = e ^ { f _ { n - 1 } ( x ) } \end{gathered}$$ How many roots does the equation $\frac { d f _ { n } ( x ) } { d x } = 0$ have in the interval $( - \infty , \infty ) ?$
(A) 1 .
(B) 3 .
(C) $n + 3$.
(D) $3n$.

The sequence of functions $f _ { 1 } ( x ) , f _ { 2 } ( x ) , f _ { 3 } ( x ) , \ldots$ is defined as follows:
$$\begin{aligned} f _ { 1 } ( x ) & = x ^ { 10 } \\ f _ { n + 1 } ( x ) & = x f _ { n } ^ { \prime } ( x ) \text { for } n \geq 1 \end{aligned}$$
where $f _ { n } ^ { \prime } ( x ) = \frac { d f _ { n } ( x ) } { d x }$
Find the value of
$$\sum _ { n = 1 } ^ { 20 } f _ { n } ( x )$$