Prove that the following functions are increasing.
$y = 2 x + s i n x f o r x ϵ R$

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Solution

$y = 2 x + sin x$

$⟹ \frac{d y}{d x} = y^{'} = 2 + cos x$

Since for all $x \in R$ , we have $- 1 \leq cos x \leq 1$

$1 \leq 2 + cos x \leq 3$

Hence, we see that $y^{'} > 0 \forall x \in R$

And, hence $y = 2 x + sin x$ is increasing for $x \in R$

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