Show that f(x)=x- is increasing for all x . - Vidyadip

Question

Show that $\text{f}(\text{x})=\text{x}-\sin\text{x}$ is increasing for all $\text{x}\in\text{R}.$

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Answer

$\text{f}(\text{x})=\text{x}-\sin\text{x}$
$\text{f}'(\text{x})=1-\cos\text{x}$
For f(x) to be increasing, we must have
$\text{f}'(\text{x})>0$
$\Rightarrow1-\cos\text{x}>0$
$\Rightarrow\text{f}'(\text{x})\geq0$ for all $\text{x}\in\text{R}$ $[\because\ \cos\text{x}\leq1]$
So, f(x) is increasing for all $\text{x}\in\text{R}.$

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