question_answer

The function \[f(x)={{x}^{3}}-3{{x}^{2}}-24x+5\]is an increasing function in the interval given below

A) \[\left( -\infty ,\,-2 \right)\cup (4,\infty )\]          

B) \[\left( -2,\infty  \right)\]

C) \[\left( -2,4 \right)\]                  

D) \[\left( -\infty ,4 \right)\]

Correct Answer: A

Solution :

\[f(x)={{x}^{3}}-3{{x}^{2}}-24x+5\]

For increasing, \[f'(x)>0,\] \[3{{x}^{2}}-6x-24>0\]

\[\Rightarrow \,{{x}^{2}}-2x-8>0\Rightarrow {{x}^{2}}-4x+2x-8>0\]

\[\Rightarrow \,\left( x+2 \right)\left( x-4 \right)>0.\]

Now, by the sign scheme for \[3{{x}^{2}}-6x-24,\]

\[\Rightarrow \,\,x\in \left( -\infty ,-2 \right)\cup \left( 4,\infty  \right)\]