`f (x) =2x -tan ^(-1) x - ln (x+ sqrt(1+ x ^(2)))`

Options

A. strictly increases `AA x in R`
B. strictly increases only in `(0,oo)`
C. strictly decreases `AA x in R`
D. strictly decreases in `(o,oo)` and strictly increases I `(-oo, 0)`

Answer

To analyze the function \( f(x) = 2x - \tan^{-1}(x) - \ln(x + \sqrt{1 + x^2}) \) and determine whether it is strictly increasing or decreasing, we will follow these steps:

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

We start by differentiating \( f(x) \):

\[ f'(x) = \frac{d}{dx}(2x) - \frac{d}{dx}(\tan^{-1}(x)) - \frac{d}{dx}(\ln(x + \sqrt{1 + x^2})) \]

Calculating each derivative:

1. The derivative of \( 2x \) is \( 2 \).
2. The derivative of \( \tan^{-1}(x) \) is \( \frac{1}{1 + x^2} \).
3. For \( \ln(x + \sqrt{1 + x^2}) \), we use the chain rule:
\[ \frac{d}{dx}(\ln(u)) = \frac{1}{u} \cdot \frac{du}{dx} \]
where \( u = x + \sqrt{1 + x^2} \). The derivative \( \frac{du}{dx} = 1 + \frac{x}{\sqrt{1 + x^2}} \).

Thus, we have:
\[ \frac{d}{dx}(\ln(x + \sqrt{1 + x^2}) = \frac{1 + \frac{x}{\sqrt{1 + x^2}}}{x + \sqrt{1 + x^2}} \]

Putting it all together, we get:
\[ f'(x) = 2 - \frac{1}{1 + x^2} - \frac{1 + \frac{x}{\sqrt{1 + x^2}}}{x + \sqrt{1 + x^2}} \]

### Step 2: Simplify \( f'(x) \)

Now we need to simplify \( f'(x) \):
\[ f'(x) = 2 - \frac{1}{1 + x^2} - \frac{1 + \frac{x}{\sqrt{1 + x^2}}}{x + \sqrt{1 + x^2}} \]

To combine these fractions, we can find a common denominator. The common denominator will be \( (1 + x^2)(x + \sqrt{1 + x^2}) \).

### Step 3: Analyze the sign of \( f'(x) \)

To determine whether \( f'(x) \) is positive or negative, we need to analyze each term.

1. The term \( 2 \) is always positive.
2. The term \( \frac{1}{1 + x^2} \) is always positive since \( 1 + x^2 > 0 \).
3. The term \( \frac{1 + \frac{x}{\sqrt{1 + x^2}}}{x + \sqrt{1 + x^2}} \) is also positive for all \( x \).

Thus, we can conclude that:
\[ f'(x) > 0 \quad \text{for all } x \in \mathbb{R} \]

### Step 4: Conclusion

Since \( f'(x) > 0 \) for all \( x \), we conclude that \( f(x) \) is a strictly increasing function for every \( x \in \mathbb{R} \).