Question

Show that f(x) = x – cos x is increasing for all x.

Answer 1

f(x) = x – cos x

∴ f'(x) = d/dx (x – cos x)

= 1 – (-sin x) 

= 1 + sin x

Now, -1 ≤ sin x ≤ 1 for all x ∈ R 

∴ -1 + 1 ≤ 1 + sin x ≤ 1 for all x ∈ R 

∴ 0 ≤ f'(x) ≤ 1 for all x ∈ R 

∴ f'(x) ≥ 0 for all x ∈ R 

∴ f is increasing for all x.