Application of Derivatives — MATHS STD 12 Science — Question
Rajasthan 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 $f(x) = -2x^3 - 9x^2 - 12x + 1$
$\Rightarrow f\ ' (x) = -6x^2 - 18x + 12$
$\Rightarrow f '(x) = -6(x^2 + 3x + 6)$
$\Rightarrow f\ ' (x) = -6(x + 1)(x + 2)$
If $f\ ' (x) = 0,$ then we get,
$\Rightarrow x = -1$ 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 $(-\infty,-2),(-1, \infty)$
$f\ ' (x) = -6(x + 1)(x + 2) < 0$
Therefore, the given function $'f\ ' $ is strictly decreasing for $x < -2$ and $x > -1$
Further, in interval $(-2, -1)$
$f\ ' (x) = -6(x + 1)(x + 2) > 0$
Therefore, the given function $(f) $ is strictly increasing for $-2 < x < -1$
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