Application of Derivatives — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
Question
Find the interval in function $-2x^3 - 9x^2 - 12x + 1$ is increasing or decreasing:
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Answer
It is given that function $f(x)=-2 x^3-9 x^2-12 x+1$
$\Rightarrow f^{\prime}(x)=-6 x^2-18 x+12$
$\Rightarrow f^{\prime}(x)=-6\left(x^2+3 x+6\right)$
$\Rightarrow f^{\prime}(x)=-6(x+1)(x+2)$
If $f^{\prime}(x)=0$, then we get,
$\Rightarrow x=-1 \text { and }-2$
So, the points $x=-1$ and $x=-2$ divides the real line into three disjoint intervals, $(-\infty,-2),(-2,-1)$ and $(-1, \infty)$
So, in intervals $(-\infty,-2),(-1, \infty)$
$f^{\prime}(x)=-6(x+1)(x+2)<0$
Therefore, the given function ' f ' is strictly decreasing for $x<-2$ and $x>-1$
Further, in interval $(-2,-1)$
$f^{\prime}(x)=-6(x+1)(x+2)>0$
Therefore, the given function ( $f$ ) is strictly increasing for $-2<x<-1$
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