Find the values of x, for which the function f(x) = x^3 + 12x^2 +… - Vidyadip

Question

Find the values of x, for which the function $f(x) = x^3 + 12x^2 + 36? + 6$ is monotonically decreasing

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Answer

$f(x) = x^3 + 12x^2 + 36? + 6$
$\therefore f′(x) = 3x^2 + 24x + 36$
$= 3(x^2 + 8x + 12)$
$= 3(x + 2)(x + 6)$
$f(x)$ is monotonically decreasing, if $f′(x) < 0$
$\therefore 3(x + 2)(x + 6) < 0$
$\therefore (x + 2)(x + 6) < 0$
$ab < 0 ⇔ a > 0$ and $b < 0$ or $a < 0$ and $b > 0$
$\therefore $ Either $x + 2 > 0$ and $x + 6 < 0$
or
$x + 2 < 0$ and $x + 6 > 0$
Case I: $x + 2 > 0$ and $x + 6 < 0$
$\therefore x > – 2$ and $x < – 6,$
which is not possible.
Case II: $x + 2 < 0$ and $x + 6 > 0$
$\therefore x < – 2$ and $x > – 6$
Thus, $f(x)$ is monotonically decreasing for $x \in (– 6, – 2)$.

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