If the function `f(x)=2tanx+(2a+1)(log)_e|secx|+(a-2)x` is increasing on `R` , then (a) `a in (1//2,\ oo)` (b) `a in (-1//2,\ 1//2)` (c) `a=1//2` (d) `a in R`

Explanation

To determine the values of \( a \) for which the function \( f(x) = 2\tan x + (2a + 1)\log_e|\sec x| + (a - 2)x \) is increasing on \( \mathbb{R} \), we need to analyze the derivative of the function.

### Step 1: Find the derivative \( f'(x) \)

The derivative of \( f(x) \) can be calculated as follows:

\[
f'(x) = \frac{d}{dx}(2\tan x) + \frac{d}{dx}((2a + 1)\log_e|\sec x|) + \frac{d}{dx}((a - 2)x)
...