Question

The set of all \( x \) for which \( \sin x \leq x \) is:

• A. \( \left(0, \frac{\pi}{2}\right) \)
• B. \( \left(\frac{\pi}{2}, \pi\right) \)
• C. \( \left(-\frac{\pi}{2}, 0\right) \)
• D. \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)

Hint:

Comparing Functions Graphically}
Remember standard inequality: \( \sin x<x \) for \( x>0 \)
Use test values: plug in \( x = 0.5, \frac{\pi}{4}, 1 \), etc.
Graphical understanding of concavity can help in verifying inequalities

Answer 1

The Correct Option is A

Consider the function \( f(x) = \sin x - x \). We analyze: - \( f(x)<0 \Rightarrow \sin x<x \Rightarrow \sin x \leq x \) - This inequality holds true in \( (0, \frac{\pi}{2}) \), as: - \( \sin x<x \) for all \( x>0 \) - \( \sin x>x \) for small negative \( x \) Hence, the valid region is: \[ \boxed{(0, \frac{\pi}{2})} \]