14. Let the trigonometric function be \( f(x) = \sin \left(\frac{k x}{5} + \frac{\pi}{3}\right) \) where \( k \neq 0 \).
(1) Write down the maximum value \( M \), the minimum value \( m \), and the smallest positive period \( T \) of \( f(x) \);
(2) Find the smallest positive integer \( k \) such that when the independent variable \( x \) varies between any two integers (including the integers themselves), the function \( f(x) \) has at least one value that is \( M \) and another value that is \( m \).
14. (1) From the conditions, we get $M=1, m=-1 . T=\frac{5 \times 2 \pi}{|k|}=\frac{10 \pi}{|k|}$.
(2) $f(x)$ in each of its periods has exactly one value that is $M$, and another value that is $m$, and the distance between any two integers is no less than 1. Therefore, to ensure that the function $f(x)$ has at least one value that is $M$ and another value that is $m$ between any two integers, the period of $f(x)$ must be no greater than 1. That is, $\frac{10 \pi}{|k|} \leqslant 1, k \geqslant 31.4$, so 32 is the smallest positive integer $k$.