Let f:

Question:

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as

$f(x)= \begin{cases}-55 x, & \text { if } x<-5 \\ 2 x^{3}-3 x^{2}-120 x, & \text { if }-5 \leq x \leq 4 \\ 2 x^{3}-3 x^{2}-36 x-336, & \text { if } x>4\end{cases}$

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

  1. (1) $(-5,-4) \cup(4, \infty)$

     

  2. (2) $(-5, \infty)$

  3. (3) $(-\infty,-5) \cup(4, \infty)$

  4. (4) $(-\infty,-5) \cup(-4, \infty)$


Correct Option: 1,

Solution:

$f(x)=\left\{\begin{array}{ccc}-55 & ; & x<-5 \\ 6\left(x^{2}-x-20\right) & ; & -54\end{array}\right.$

$f(x)=\left\{\begin{array}{lcc}6(x-5)(x+4) & ; & -54\end{array}\right.$

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

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