Question

Let the function f : R → R be defined by f(x) = 2x + cos x, then f(x)
(a) has a minimum at x = π (b) has a maximum at x = 0
(c) is a decreasing function (d) is an increasing function

Answer 1

The given function is f(x) = 2x + cosx.

f(x) = 2x + cosx

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=2-\sin x$

We know

−1 ≤ sinx ≤ 1

$\therefore f\text{'}\left(x\right)=2-\sin x>0$ ∀ x ∈ R

⇒ f(x) is an increasing function for all x ∈ R

Thus, the function f : R → R defined by f(x) = 2x + cosx is an increasing function.

Hence, the correct answer is option (d).