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Question
Show that f(x) = x – cos x is increasing for all x.
Solution
f(x) = x – cos x
∴ f′(x) = 1 + sin x
Note that –1 ≤ sin x ≤ 1, ∀x
∴ –1 + 1 ≤ 1 + sin x ≤ 1 + 1, ∀x
∴ 0 ≤ 1 + sin x ≤ 2, ∀x
i.e., f′(x) ≥ 0 for all x.
Hence, f(x) is increasing for all x
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