Mark the correct alternative in the following:
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 $f^{\prime}(x)>0$

$\Rightarrow 9 x^{8}+21 x^{6}>0$

$\Rightarrow x \in R$

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