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Show that f (x) = 2x + cot-1 x + log [√(1 + x2) – x] is increasing in R.

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Given: f (x) = 2x + cot-1 x + log [√(1 + x2) – x]

To show: the given function is increasing in R.

Applying the sum rule to last part we get

Now adding by taking the LCM, we get

Now for any real value of x, the above value of f(x) is greater than or equal to zero

And we know, if f’(x)≥0, then f(x) is increasing function.

Hence the given function is increasing function in R.

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