Question

Consider the function
$f\left(x\right)=-2x^3-9x^2-12x+1$
The function $f\left(x\right)$ is a decreasing function in the interval

• A. $(-2,-1)$
• B. $(-\infty ,-2)$ only
• C. $(-1,\infty )$ only
• D. $(-\infty ,-2)\cup (-1,\infty )$

Answer 1

Answer: (D)

$(-\infty ,-2)\cup (-1,\infty )$

Solution:

Given that:

$f\left(x\right)=-2x^3-9x^2-12x+1$

On differentiating both sides w.r.t x. we get

$f^{\prime }\left(x\right)=-6x^2-18x-12$

For $f\left(x\right)$ to be decreasing $f^{\prime }\left(x\right)<0$

$\therefore -6x^2-18x-12<0$

$⟹x^2+3x+2>0⟹(x+1)(x+2)>0$

$\therefore x\in (-\infty ,-2)\cup (-1,\infty )$

Hence, D is the correct option.