15. Given functions $f ( x ) = 2 ^ { x }$ and $g ( x ) = x ^ { 2 } + a x$ (where $a \in \mathbb{R}$). For unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, let $m = \frac { f \left( x _ { 1 } \right) - f \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$ and $n = \frac { g \left( x _ { 1 } \right) - g \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$. The following propositions are given: (1) For any unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, we have $m > 0$; (2) For any $a$ and any unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, we have $n > 0$; (3) For any $a$, there exist unequal real numbers $x _ { 1 }$ and $x _ { 2 }$ such that $m = n$; (4) For any $a$, there exist unequal real numbers $x _ { 1 }$ and $x _ { 2 }$ such that $m = - n$. The true propositions are $\_\_\_\_$ (write out the numbers of all true propositions).
III. Solution Questions
15. Given functions $f ( x ) = 2 ^ { x }$ and $g ( x ) = x ^ { 2 } + a x$ (where $a \in \mathbb{R}$). For unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, let $m = \frac { f \left( x _ { 1 } \right) - f \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$ and $n = \frac { g \left( x _ { 1 } \right) - g \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$.
The following propositions are given:\\
(1) For any unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, we have $m > 0$;\\
(2) For any $a$ and any unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, we have $n > 0$;\\
(3) For any $a$, there exist unequal real numbers $x _ { 1 }$ and $x _ { 2 }$ such that $m = n$;\\
(4) For any $a$, there exist unequal real numbers $x _ { 1 }$ and $x _ { 2 }$ such that $m = - n$. The true propositions are $\_\_\_\_$ (write out the numbers of all true propositions).
\section*{III. Solution Questions}