UPSC NDA Math Model Paper 2

Let f(x) = x - sin x
Now, differentiating both sides w.r.t x
⇒ f’(x) = 1 - cos x
We know, -1 ≤ cos x ≤ 1 for all x > 0
⇒ 1 ≥ -cos x ≥ -1
⇒ 2 ≥ 1 - cos x ≥ 0
∴ f’(x) ≥ 0 for all x > 0
Hence, f(x) is an increasing function for all x > 0.
And since, f(x) is an increasing function for all x > 0, then x > sin x for all x > 0.
Hence, both statement I & II are correct and statement II is correct explanation of statement I.