16. If the minimum value of the function $f ( x ) = | x + 1 | + 2 | x - a |$ is 5, then the real number $a = $ $\_\_\_\_$ . III. Solution Questions: This section contains 6 questions, for a total of 75 points. Show your work, proofs, or calculation steps. (17) (This question is worth 13 points: part (I) is worth 5 points, part (II) is worth 8 points) Eating zongzi on Dragon Boat Festival is a traditional custom in China. A plate contains 10 zongzi: 2 with red bean paste, 3 with meat, and 5 plain. The three types of zongzi look identical. Three zongzi are randomly selected. (I) Find the probability that one zongzi of each type is selected. (II) Let $X$ denote the number of red bean paste zongzi selected. Find the probability distribution and mathematical expectation of $X$. (18) (This question is worth 13 points: part (I) is worth 7 points, part (II) is worth 6 points) Given the function $f ( x ) = \sin \left( \frac { \pi } { 2 } - x \right) \sin x - \sqrt { 3 } \cos ^ { 2 } x$ (I) Find the minimum positive period and maximum value of $f ( x )$. (II) Discuss the monotonicity of $f ( x )$ on $\left[ \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } \right]$. (19) (This question is worth 13 points: part (I) is worth 4 points, part (II) is worth 9 points) As shown in question (19), in the triangular pyramid $P - A B C$, $P C \perp$ plane $A B C$, $P C = 3$, $\angle A C B = \frac { \pi } { 2 }$. Points $D$ and $E$ are on segments $A B$ and $B C$ respectively, with $C D = D E = \sqrt { 2 }$, $C E = 2 E B = 2$. (I) Prove that $D E \perp$ plane $P C D$. (II) Find the cosine of the dihedral angle $A - P D - C$. (20) (This question is worth 12 points: part (I) is worth 7 points, part (II) is worth 5 points) Let the function $f ( x ) = \frac { 3 x ^ { 2 } + a x } { e ^ { x } } ( a \in R )$. (I) If $f ( x )$ has an extremum at $x = 0$, determine the value of $a$ and find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$. (II) If $f ( x )$ is decreasing on $[ 3 , + \infty )$, find the range of values for $a$. (21) (This question is worth 12 points: part (I) is worth 5 points, part (II) is worth 7 points)
[Figure]
Question (19) Figure
As shown in question (21), the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ {

What is the minimum value of the function $| x - 3 | + | x + 2 | + | x + 1 | + | x |$ for real $x$?
(A) 3
(B) 5
(C) 6
(D) 8

Consider the expression in $x$
$$P = | x - 1 | + | x - 2 | + | x - a | .$$
We are to find the range of real numbers $a$ such that the value of $P$ is minimized at $x = a$.
First, let us note that the inequality
$$| x - 1 | + | x - 2 | + | x - a | \geqq | x - 1 | + | x - 2 |$$
always holds, and is an equality in the case $x = a$.
When we set
$$y = | x - 1 | + | x - 2 | ,$$
we have
$$y = \begin{cases} - \mathbf { A } & x + \mathbf { B } \\ \mathbf { D } & ( x < \mathbf { C } ) \\ \mathbf { F } & ( \mathbf { C } \leqq x \leqq \mathbf { E } ) \\ \mathbf { G } & ( \mathbf { E } < x ) . \end{cases}$$
When we consider the graph of (1), we see that the minimum value of $y$ is $\mathbf { H }$ and $y$ takes the value $\mathbf{H}$ at every $x$ satisfying $\mathbf{I} \leqq x \leqq \mathbf{J}$.
Thus, for every $a$ satisfying $\mathbf{K} \leqq a \leqq \mathbf{L}$, the value of $P$ is minimized at $x = a$ and its value there is $\mathbf{M}$.