$\text{ Let }f\left(x\right)=\left\{\begin{array}{r}x{e}^{ax};\hspace{0.25em} \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\le 0\\ x+a{x}^2-x^3;\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x>0\end{array} \text{where} a\right.\text{ ' is a positive constant. Find }$$\text{ the interval in which }{f}^{\mathrm{\prime }}\left(x\right)\text{ is increasing. }$

$f^{\mathrm{\prime }}\left(x\right)=\left\{\begin{array}{l}{e}^{ax}+ax{e}^{ax};x\le 0\\ 1+2ax-3x^2;x>0\end{array}\right.$

$\begin{array}{l}{f}^{\prime \prime }\left(x\right)=\left\{\begin{array}{ll}2a{e}^{ax}+a^{2}x{e}^{ax}& ;x\le 0\\ 2a-6x& ;x>0\end{array}\right.\\ \text{ For }{f}^{\mathrm{\prime }}\left(x\right)\text{ to be an increasing function }{f}^{\prime \prime }\left(x\right)\ge 0\\ \text{ i.e., }a{e}^{ax}(2+ax)\ge 0\Rightarrow x\ge -\frac{2}{a}\end{array}$

$2a-6x\ge 0\Rightarrow x\le \frac{a}{3}$

$\therefore \text{ The int erval inwhich }{f}^{\mathrm{\prime }}\left(x\right)\text{ is increasing is }\left[-\frac{2}{a},\frac{a}{3}\right]$