Find 'a' for which f(x)=a(x+ )+a is increasing on R. - Vidyadip

Question

Find 'a' for which $\text{f}(\text{x})=\text{a}(\text{x}+\sin\text{x})+\text{a}$ is increasing on R.

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Answer

$\text{f}(\text{x})=\text{a}(\text{x}+\sin\text{x})+\text{a}$
$\text{f}'(\text{x})=\text{a}(1+\cos\text{x})$
For f(x) to be increasing, we must have
$\text{f}'(\text{x})>0$
$\Rightarrow\text{a}(1+\cos\text{x})>0\ ....(1)$
We know,
$-1\leq\cos\text{x}\leq1,\forall\ \text{x}\in\text{R}$
$\Rightarrow0\leq(1+\cos\text{x})\leq2,\forall\ \text{x}\in\text{R}$
$\therefore\ \text{a}>0$ [From eq. (1)]
$\Rightarrow\text{a}\in(0,\infty)$

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