Question

The function $f(x) = 5 + 36x + 3x^2 - 2x^3$ is decreasing in the interval

• A. $(-2, 3)$
• B. $(2, -3)$
• C. $(-2, -3)$
• D. $(2, 3)$

Answer 1

Answer: (A)

Given , $f\left(x\right) = 5+36x+3x^{2} =2x^{3}$
$ \Rightarrow f'\left(x\right) = 36 +6x-6x^{2} $
For increasing or decreasing; $ f'(x) = 0 $
$\Rightarrow 6+x-x^{2} =0 \Rightarrow x^{2} -x-6 =0 $
$\Rightarrow \left(x-3\right)\left(x+2\right) = 0 \Rightarrow x=3,-2$
Intervals:
$-\infty < x < -2 f'\left(x\right) =\left(-ve\right)\left(-ve\right)= \left(+ve\right)$, Increasing $ -2 < x < 0$
$ f'\left(x\right) =\left(-ve\right)\left(+ve\right)=\left(-ve\right), $ Decreasing $ 0 < x < 3 $
$f'\left(x\right)=\left(-ve\right)\left(+ve\right)=\left(-ve\right), $ Decreasing $3< x <\infty $
$f'\left(x\right) =\left(+ve\right)\left(+ve\right) =\left(+ve\right)$ Increasing
$ \therefore $ The interval in which $f(x)$ is decreasing is $(-2, 3)$