Question:
Let the function $f: R \rightarrow R$ be defined by $f(x)=2 x+\cos x$, then $f(x)$
(a) has a minimum at $x=\pi$ (b) has a maximum at $x=0$
(c) is a decreasing function (d) is an increasing function
Solution:
The given function is $f(x)=2 x+\cos x$.
$f(x)=2 x+\cos x$
Differentiating both sides with respect to $x$, we get
$f^{\prime}(x)=2-\sin x$
We know
$-1 \leq \sin x \leq 1$
$\therefore f^{\prime}(x)=2-\sin x>0 \forall x \in \mathrm{R}$
$\Rightarrow f(x)$ is an increasing function for all $x \in \mathrm{R}$'
Thus, the function $f: \mathrm{R} \rightarrow \mathrm{R}$ defined by $f(x)=2 x+\cos x$ is an increasing function.
Hence, the correct answer is option (d).