Function f(x) = 2x^3 - 9x^2 + 12x + 29 is monotonically decreasin… - Vidyadip

MCQ

Function $f(x) = 2x^3 - 9x^2 + 12x + 29$ is monotonically decreasing when:

A
$x < 2$
B
$x > 2$
C
$x > 3$
✓
$1 < x < 2$

✓

Answer

Correct option: D.

$1 < x < 2$

$f(x) = 2x^3 - 9x^2 + 12x + 29$
$\Rightarrow f'(x) = 6x^2 - 18x + 12$
$\Rightarrow f'(x) = 6(x^2 - 3x + 2)$
$\Rightarrow f'(x) = 6(x - 1)(x - 2)$
For $f(x)$ to be decreasing, we must have
$f'(x) < 0$
$\Rightarrow 6(x - 1)(x - 2) < 0$
$\Rightarrow (x - 1)(x - 2) < 0$
$[$Since, $6 > 0, 6(x - 1)(x - 2) < 0 $
$\Rightarrow (x - 1)(x - 2) < 0]$
$\Rightarrow 1 < x < 2$
So, $f(x)$ is decreasing for $1 < x < 2.$

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