Question:
Mark the correct alternative in the following:
The function $f(x)=x^{9}+3 x^{7}+64$ is increasing on
A. $\mathrm{R}$
B. $(-\infty, 0)$
C. $(0, \infty)$
D. $R_{0}$
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:-
$f(x)=x^{9}+3 x^{7}+64$
$\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=9 \mathrm{x}^{8}+21 \mathrm{x}^{6}=\mathrm{f}^{\prime}(\mathrm{x})$
For increasing $f^{\prime}(x)>0$
$\Rightarrow 9 x^{8}+21 x^{6}>0$
$\Rightarrow x \in R$