#### Question

Show that f(x) = x − sin x is increasing for all x ∈ R ?

#### Solution

\[f\left( x \right) = x - \sin x\]

\[f'\left( x \right) = 1 - \cos x\]

\[\text { For f(x) to be increasing, we must have}\]

\[f'\left( x \right) > 0\]

\[ \Rightarrow 1 - \cos x > 0\]

\[ \Rightarrow f'(x) \geqslant 0 \text { for all } x \in R \left[ \because Cos x \leqslant 1 \right]\]

\[\text { So, f(x) is increasing for all } x \in R . \]

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Solution Show that F(X) = X − Sin X is Increasing for All X ∈ R ? Concept: Increasing and Decreasing Functions.