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Question
Show that f(x) = tan−1 x − x is a decreasing function on R ?
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Solution
\[f\left( x \right) = \tan^{- 1} x - x\]
\[f'\left( x \right) = \frac{1}{1 + x^2} - 1\]
\[ = \frac{1 - 1 - x^2}{1 + x^2}\]
\[ = \frac{- x^2}{1 + x^2}\]
\[\text { We know,}\]
\[ x^2 \geq 0, 1+ x^2 >0, \forall x \in R\]
\[ \therefore \frac{- x^2}{1 + x^2} < 0, \forall x \in R\]
\[ \Rightarrow f'\left( x \right) < 0, \forall x \in R\]
\[\text { So,}f\left( x \right) \text { is decreasing on R }.\]
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