The function `f(x)=(tan^(-1)x)^3-(cot^(-1)x)^2+tan^(-1)x+2` is decreasing `AAx in R` increasing `AAx in R` bounded (d) Many one functions

Options

A. Decreasing `AA x in R`
B. Increasing `AA x in R`
C. Many one function
D. None of these

Answer

To determine the nature of the function \( f(x) = (\tan^{-1} x)^3 - (\cot^{-1} x)^2 + \tan^{-1} x + 2 \), we will follow these steps:

### Step 1: Differentiate the function
We need to find \( f'(x) \) to analyze whether the function is increasing or decreasing.

Using the chain rule and the derivatives of the inverse trigonometric functions, we differentiate each term:
- The derivative of \( (\tan^{-1} x)^3 \) is \( 3(\tan^{-1} x)^2 \cdot \frac{1}{1 + x^2} \).
- The derivative of \( -(\cot^{-1} x)^2 \) is \( -2(\cot^{-1} x) \cdot \left(-\frac{1}{1 + x^2}\right) = \frac{2(\cot^{-1} x)}{1 + x^2} \).
- The derivative of \( \tan^{-1} x \) is \( \frac{1}{1 + x^2} \).
- The derivative of the constant \( 2 \) is \( 0 \).

So, we can write:
\[ f'(x) = 3(\tan^{-1} x)^2 \cdot \frac{1}{1 + x^2} + \frac{2(\cot^{-1} x)}{1 + x^2} + \frac{1}{1 + x^2} \]

### Step 2: Simplify the derivative
Now, we can factor out \( \frac{1}{1 + x^2} \):
\[ f'(x) = \frac{1}{1 + x^2} \left( 3(\tan^{-1} x)^2 + 2(\cot^{-1} x) + 1 \right) \]

### Step 3: Analyze the sign of \( f'(x) \)
The term \( 1 + x^2 \) is always positive for all \( x \in \mathbb{R} \).

Next, we need to analyze the expression \( 3(\tan^{-1} x)^2 + 2(\cot^{-1} x) + 1 \):
- \( \tan^{-1} x \) ranges from \( 0 \) to \( \frac{\pi}{2} \) as \( x \) goes from \( 0 \) to \( \infty \), and \( \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \) also ranges from \( \frac{\pi}{2} \) to \( 0 \).
- Both \( (\tan^{-1} x)^2 \) and \( \cot^{-1} x \) are non-negative.

Since both components are non-negative, we can conclude that \( 3(\tan^{-1} x)^2 + 2(\cot^{-1} x) + 1 \) is also positive for all \( x \).

### Step 4: Conclusion
Since \( f'(x) > 0 \) for all \( x \in \mathbb{R} \), we conclude that the function \( f(x) \) is increasing for all \( x \in \mathbb{R} \).

### Final Answer
The function \( f(x) \) is **increasing** for all \( x \in \mathbb{R}**.

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