Question

If $f: \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = 2x - \sin x$, then $f$ is:

• A. a decreasing function
• B. an increasing function
• C. maximum at $x = \frac{\pi}{2}$
• D. maximum at $x = 0$

Hint:

If the derivative of a function is always positive, the function is always increasing.

Answer 1

The Correct Option is B

The function is $f(x) = 2x - \sin x$. To determine its monotonicity, we find the derivative: \[ f'(x) = \frac{d}{dx}(2x - \sin x) = 2 - \cos x \] Given that the range of $\cos x$ is $[-1, 1]$, it follows that $f'(x) = 2 - \cos x \geq 1$. As the derivative is consistently positive, the function $f(x)$ is always increasing.