For the function f(x)=x cos 1x,  x≥1,

We have f(x)=x cos 1x, x≥1∴  f'(x)=cos 1x +1x sin1xlimx→∞ f'(x)=cos 0 +(0) × (some  finite  value)⇒limx→∞ f'(x)=1Also  f''(x)=1x2 sin1x −1x2sin 1x −1x3cos 1x⇒f''(x)=−1x3 cos 1x <0,  ∀ x∈[1,∞)⇒f'(x)  is  strictly  decreasing  in  [1,∞)∴  f'(x)>limx→∞f'(x)⇒  f(x+2)−f(x)(x+2) −x  >1 ⇒  f(x+2) −f(x)>2