Question
Find the intervals in which the function $f(x)=-2 x^3$$-9 x^2-12 x+1$is (i) Strictly increasing (ii) strictly decreasing.
✓
Answer
$f(x)=-2 x^3-9 x^2-12 x+1$
Now, $f^{\prime}(x)=-6 x^2-18 x-12$
$=-6\left[x^2+3 x+2\right]$
$=-6\left[x^2+2 x+x+2\right]$
$f^{\prime}(x)=-6(x+1)(x+2)$
$\Rightarrow$ Intervals are $(-\infty,-2),(-2,-1)$ and $(-1, \infty)$
Getting $f^{\prime}(x)>0$ in $(-2,-1)$ and $f^{\prime}(x)<0$ in $(-\infty,-2)$$\cup(-1, \infty)$
$\Rightarrow f(x)$ is strictly increasing in $(-2,-1)$
and strictly decreasing in $(-\infty,-2) \cup(-1, \infty)$
Now, $f^{\prime}(x)=-6 x^2-18 x-12$
$=-6\left[x^2+3 x+2\right]$
$=-6\left[x^2+2 x+x+2\right]$
$f^{\prime}(x)=-6(x+1)(x+2)$
$\Rightarrow$ Intervals are $(-\infty,-2),(-2,-1)$ and $(-1, \infty)$
Getting $f^{\prime}(x)>0$ in $(-2,-1)$ and $f^{\prime}(x)<0$ in $(-\infty,-2)$$\cup(-1, \infty)$
$\Rightarrow f(x)$ is strictly increasing in $(-2,-1)$
and strictly decreasing in $(-\infty,-2) \cup(-1, \infty)$
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