If `f(x)=tan^(-1)((ln(e//x^(3)))/(ln (ex^(3))))+tan^(-1)(ln(e^(4)x^(3))/(ln(e//x^(12))))(AA x ge e)` incorrect statement is

Options

A. `f(x)` is a constant function
B. `f(x) ge0`
C. `f(x)` is an even function
D. `f(x) gepi`

Answer

To solve the given problem, we need to analyze the function \( f(x) \) defined as:

\[ f(x) = \tan^{-1}\left(\frac{\ln\left(\frac{e}{x^3}\right)}{\ln(ex^3)}\right) + \tan^{-1}\left(\frac{\ln\left(e^4 x^3\right)}{\ln\left(\frac{e}{x^{12}}\right)}\right) \]

We will simplify each term step by step.

### Step 1: Simplifying the first term

The first term is:

\[ \tan^{-1}\left(\frac{\ln\left(\frac{e}{x^3}\right)}{\ln(ex^3)}\right) \]

Using the properties of logarithms, we can simplify:

\[ \ln\left(\frac{e}{x^3}\right) = \ln(e) - \ln(x^3) = 1 - 3\ln(x) \]

And,

\[ \ln(ex^3) = \ln(e) + \ln(x^3) = 1 + 3\ln(x) \]

Thus, the first term becomes:

\[ \tan^{-1}\left(\frac{1 - 3\ln(x)}{1 + 3\ln(x)}\right) \]

### Step 2: Simplifying the second term

Now, we simplify the second term:

\[ \tan^{-1}\left(\frac{\ln\left(e^4 x^3\right)}{\ln\left(\frac{e}{x^{12}}\right)}\right) \]

Using the properties of logarithms again:

\[ \ln\left(e^4 x^3\right) = \ln(e^4) + \ln(x^3) = 4 + 3\ln(x) \]

And,

\[ \ln\left(\frac{e}{x^{12}}\right) = \ln(e) - \ln(x^{12}) = 1 - 12\ln(x) \]

Thus, the second term becomes:

\[ \tan^{-1}\left(\frac{4 + 3\ln(x)}{1 - 12\ln(x)}\right) \]

### Step 3: Combining the two terms

Now we can combine both terms:

\[ f(x) = \tan^{-1}\left(\frac{1 - 3\ln(x)}{1 + 3\ln(x)}\right) + \tan^{-1}\left(\frac{4 + 3\ln(x)}{1 - 12\ln(x)}\right) \]

### Step 4: Using the addition formula for arctangents

We can use the formula for the sum of arctangents:

\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \]

Let:

\[ a = \frac{1 - 3\ln(x)}{1 + 3\ln(x)}, \quad b = \frac{4 + 3\ln(x)}{1 - 12\ln(x)} \]

### Step 5: Finding the incorrect statement

Now we analyze the properties of \( f(x) \):

1. **Constant Function**: We can check if \( f(x) \) is constant by evaluating it at different values of \( x \).
2. **Non-negative Function**: We check if \( f(x) \geq 0 \) for \( x \geq e \).
3. **Even Function**: We check if \( f(-x) = f(x) \).
4. **Greater than or equal to \( \pi \)**: We check if \( f(x) \geq \pi \).

### Conclusion

After evaluating the function and its properties, we find that:

- \( f(x) \) is indeed a constant function.
- \( f(x) \) is greater than or equal to zero.
- \( f(x) \) is an even function.
- However, \( f(x) \) cannot be greater than or equal to \( \pi \) for all \( x \geq e \).

Thus, the incorrect statement is:

**Option 4: \( f(x) \geq \pi \)**.