Question:
Mark the correct alternative in the following:
Function $f(x)=x^{3}-27 x+5$ is monotonically increasing when
A. $x<-3$
B. $|x|>3$
C. $x \leq-3$
D. $|x| \geq 3$
Solution:
Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$
Given:-
$f(x)=x^{3}-27 x+5$
$\frac{d(f(x))}{d x}=3 x^{2}-27=f^{\prime}(x)$
for increasing function $f^{\prime}(x)>0$
$3 x^{2}-27>0$
$\Rightarrow(x+3)(x-3)>0$
$\Rightarrow|x|>3$