21. (This question is worth 13 points) Given $\mathrm { a } > 0$, the function $\mathrm { f } ( \mathrm { x } ) = \mathrm { a } e ^ { x } \cos x$ for $\mathrm { x } \in [ 0 , + \infty )$. Let $x _ { n }$ denote the $n$-th (where $n \in \mathbb { N } ^ { * }$) extremum point of $f ( x )$ in increasing order. (I) Prove that: the sequence $\left\{ f \left( \mathrm { x } _ { \mathrm { n } } \right) \right\}$ is a geometric sequence; (II) If for all $n \in \mathbb { N } ^ { * }$, the inequality $x _ { n } \leq \left| f \left( x _ { n } \right) \right|$ always holds, find the range of $a$.
21. (This question is worth 13 points)\\
Given $\mathrm { a } > 0$, the function $\mathrm { f } ( \mathrm { x } ) = \mathrm { a } e ^ { x } \cos x$ for $\mathrm { x } \in [ 0 , + \infty )$. Let $x _ { n }$ denote the $n$-th (where $n \in \mathbb { N } ^ { * }$) extremum point of $f ( x )$ in increasing order.\\
(I) Prove that: the sequence $\left\{ f \left( \mathrm { x } _ { \mathrm { n } } \right) \right\}$ is a geometric sequence;\\
(II) If for all $n \in \mathbb { N } ^ { * }$, the inequality $x _ { n } \leq \left| f \left( x _ { n } \right) \right|$ always holds, find the range of $a$.