If the domain of `f (x) =1/picos ^(-1)[log _(3) ((x^(2))/(3))]` where, `x gt 0` is [a,b] and the range of `f (x)` is [c,d], then :

Options

A. a,b are the roots of the equation `x ^(4) -3x ^(4) -3xc ^(3) -x+3 =0`
B. a,b are the roots of the equatin `x ^(4) -x ^(3) +x^(2)-2x+1 =0`
C. `a ^(3) +d ^(3) =1`
D. `a ^(2) +b^(2) +c^(2)=11`

Answer

To solve the problem, we need to find the domain and range of the function \( f(x) = \frac{1}{\pi} \cos^{-1} \left( \log_3 \left( \frac{x^2}{3} \right) \right) \) where \( x > 0 \).

### Step 1: Determine the domain of \( f(x) \)

1. **Logarithm Condition**: The expression inside the logarithm must be positive. Since \( x > 0 \), we have:
\[ \frac{x^2}{3} > 0 \quad \text{(which is always true for } x > 0\text{)} \]

2. **Cosine Inverse Condition**: The argument of the \( \cos^{-1} \) function must be in the interval \([-1, 1]\):
\[ -1 \leq \log_3 \left( \frac{x^2}{3} \right) \leq 1 \]

3. **Convert Logarithmic Inequalities**:
- For \( \log_3 \left( \frac{x^2}{3} \right) \leq 1 \):
\[ \frac{x^2}{3} \leq 3 \implies x^2 \leq 9 \implies x \leq 3 \]
- For \( \log_3 \left( \frac{x^2}{3} \right) \geq -1 \):
\[ \frac{x^2}{3} \geq \frac{1}{3} \implies x^2 \geq 1 \implies x \geq 1 \]

4. **Combine the Conditions**: From the inequalities \( x \geq 1 \) and \( x \leq 3 \), we find:
\[ 1 \leq x \leq 3 \]
Thus, the domain of \( f(x) \) is:
\[ [1, 3] \]

### Step 2: Determine the range of \( f(x) \)

1. **Evaluate the limits of \( f(x) \)**:
- At \( x = 1 \):
\[ f(1) = \frac{1}{\pi} \cos^{-1} \left( \log_3 \left( \frac{1^2}{3} \right) \right) = \frac{1}{\pi} \cos^{-1} \left( \log_3 \left( \frac{1}{3} \right) \right) = \frac{1}{\pi} \cos^{-1}(-1) = \frac{1}{\pi} \cdot \pi = 1 \]

- At \( x = 3 \):
\[ f(3) = \frac{1}{\pi} \cos^{-1} \left( \log_3 \left( \frac{3^2}{3} \right) \right) = \frac{1}{\pi} \cos^{-1} \left( \log_3(3) \right) = \frac{1}{\pi} \cos^{-1}(1) = \frac{1}{\pi} \cdot 0 = 0 \]

2. **Range of \( f(x) \)**: Since \( f(x) \) decreases from 1 to 0 as \( x \) goes from 1 to 3, the range of \( f(x) \) is:
\[ [0, 1] \]

### Final Result

- The domain of \( f(x) \) is \([1, 3]\) (where \( a = 1 \) and \( b = 3 \)).
- The range of \( f(x) \) is \([0, 1]\) (where \( c = 0 \) and \( d = 1 \)).