The inequality `x^(2)-3x gt tan^(-1) x` is ture in

Options

A. `(4, infty)`
B. `(infty,0)`
C. (0, 1)
D. (0, 2)

Answer

To solve the inequality \( x^2 - 3x > \tan^{-1}(x) \), we will follow these steps:

### Step 1: Rearrange the Inequality
We can rewrite the inequality as:
\[ x^2 - 3x - \tan^{-1}(x) > 0 \]
Let \( f(x) = x^2 - 3x - \tan^{-1}(x) \).

### Step 2: Find the Derivative
Next, we need to find the derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(x^2 - 3x) - \frac{d}{dx}(\tan^{-1}(x)) \]
Calculating the derivatives:
\[ f'(x) = 2x - 3 - \frac{1}{1+x^2} \]

### Step 3: Combine the Terms
To analyze the sign of \( f'(x) \), we can combine the terms over a common denominator:
\[ f'(x) = \frac{(2x - 3)(1 + x^2) - 1}{1 + x^2} \]
This simplifies to:
\[ f'(x) = \frac{2x^3 - 3x^2 + 2x - 4}{1 + x^2} \]

### Step 4: Analyze the Critical Points
Now, we need to find the critical points by setting the numerator equal to zero:
\[ 2x^3 - 3x^2 + 2x - 4 = 0 \]
We can use numerical methods or graphing to find the roots of this cubic equation.

### Step 5: Determine the Nature of the Function
To determine where \( f'(x) > 0 \) or \( f'(x) < 0 \), we can test intervals around the roots found in the previous step.

### Step 6: Evaluate the Function at Specific Points
Evaluate \( f(x) \) at specific points to determine where the function is positive:
- Calculate \( f(0) \):
\[ f(0) = 0^2 - 3(0) - \tan^{-1}(0) = 0 - 0 = 0 \]
- Calculate \( f(1) \):
\[ f(1) = 1^2 - 3(1) - \tan^{-1}(1) = 1 - 3 - \frac{\pi}{4} < 0 \]
- Calculate \( f(2) \):
\[ f(2) = 2^2 - 3(2) - \tan^{-1}(2) = 4 - 6 - \tan^{-1}(2) < 0 \]
- Calculate \( f(3) \):
\[ f(3) = 3^2 - 3(3) - \tan^{-1}(3) = 9 - 9 - \tan^{-1}(3) < 0 \]
- Calculate \( f(4) \):
\[ f(4) = 4^2 - 3(4) - \tan^{-1}(4) = 16 - 12 - \tan^{-1}(4) > 0 \]

### Step 7: Conclusion
From the evaluations, we can conclude that the function \( f(x) > 0 \) for \( x > 4 \). Therefore, the solution to the inequality \( x^2 - 3x > \tan^{-1}(x) \) is:
\[ x \in (4, \infty) \]