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