Question:
Mark the correct alternative in the following:
Function $f(x)=2 x^{3}-9 x^{2}+12 x+29$ is monotonically decreasing when
A. $x<2$
B. $x>2$
C. $x>3$
D. $1
Solution:
Formula:- (i) The necessary and sufficient condition for differentiable function defined on ( $a, b$ ) to be strictly decreasing on $(a, b)$ is that $f^{\prime}(x)<0$ for all $x \in(a, b)$
Given:-
$f(x)=2 x^{3}-9 x^{2}+12 x+29$
$\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=\mathrm{f}^{\prime}(\mathrm{x})=6(\mathrm{x}-1)(\mathrm{x}-2)$
for decreasing function $f^{\prime}(x)<0$
$f^{\prime}(x)<0$
$\Rightarrow 6(x-1)(x-2)<0$
$\Rightarrow 1