Question

Let $f:R\to R$ be defined as
$f\left(x\right)=\{\begin{array}{cc}-55x,& \text{if}x<-5\\ 2x^3-3x^2-120x,& \text{if}-5\le x\le 4\\ 2x^3-3x^2-36x-336,& \text{if}x>4\end{array}$

Let $A=\{x\in R:f$ is increasing$\}.$ Then $A$ is equal to

• A. $(-5,-4)\cup (4,\infty )$
• B. $(-5,\infty )$
• C. $(-\infty ,-5)\cup (4,\infty )$
• D. $(-\infty ,-5)\cup (-4,\infty )$

Answer 1

The correct option is A $(-5,-4)\cup (4,\infty )$

$f^{\prime }\left(x\right)=\{\begin{array}{ccc}-55& ;& x<-5\\ 6(x^2-x-20)& ;& -5\le x\le 4\\ 6(x^2-x-6)& ;& x>4\end{array}$

$\Rightarrow f^{\prime }\left(x\right)=\{\begin{array}{ccc}-55& ;& x<-5\\ 6(x-5)(x+4)& ;& -5\le x\le 4\\ 6(x-3)(x+2)& ;& x>4\end{array}$

Hence, $f\left(x\right)$ is monotonically increasing in $(-5,-4)\cup (4,\infty )$