Let the function

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).

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