$\text{ The function }$$f\left(x\right)=x^3-3x^2-24x+5\text{ is an }$increasing function in the interval

$\begin{array}{l}f\left(x\right)=x^3-3x^2-24x+5\\ \text{ For increasing, }{f}^{\text{'}}\left(x\right)>0,3x^2-6x-24>0\\ \Rightarrow x^2-2x-8>0\Rightarrow x^2-4x+2x-8>0\\ \Rightarrow (x+2)(x-4)>0\text{ . }\\ \text{ Now, by the sign scheme for }3x^2-6x-24\text{ , }\end{array}$

$\Rightarrow x\in (-\infty ,-2)\cup (4,\infty )$