`theta` is said to be well behaved if it lies in interval `[0,(pi)/(2)].` They are intelligent if they make domain of `f +g and g` equal. The vlaue of `theta` for which `h (theta)` is defined are handosome. Let
`f (x)= sqrt(thetax ^(2) -2 (theta^(2) -3) x-12theta,) g (x)=ln (x^(2) -49),`
`h (theta) ln [int_(0)^(theta) 4 cos ^(2)t dt - theta ^(2)],` where `theta` is in radians.
Complete set of vlaues of `theta` which are well behaved as well as intellignent is:

Options

A. `[(3)/(4),(pi)/(2)]`
B. `[(3)/(5), (7)/(8)]`
C. `[(5)/(6), (pi)/(2)]`
D. `[(6)/(7) , (pi)/(2)]`

Answer

To solve the problem, we need to find the values of \(\theta\) that are both "well behaved" and "intelligent". Let's break down the steps involved in the solution.

### Step 1: Define the Conditions for \(\theta\)

1. **Well Behaved Condition**:
\[ \theta \in [0, \frac{\pi}{2}] \]

2. **Intelligent Condition**:
The domain of \(f + g\) should be equal to the domain of \(g\).

### Step 2: Determine the Domain of \(g(x)\)

The function \(g(x) = \ln(x^2 - 49)\) is defined when:
\[ x^2 - 49 > 0 \implies |x| > 7 \]
Thus, the domain of \(g\) is:
\[ (-\infty, -7) \cup (7, \infty) \]

### Step 3: Determine the Domain of \(f(x)\)

The function \(f(x) = \sqrt{\theta x^2 - 2(\theta^2 - 3)x - 12\theta}\) is defined when:
\[ \theta x^2 - 2(\theta^2 - 3)x - 12\theta \geq 0 \]
This is a quadratic inequality in \(x\). The quadratic will be defined for values of \(x\) outside its roots.

### Step 4: Find the Roots of the Quadratic

To find the roots of the quadratic equation:
\[ \theta x^2 - 2(\theta^2 - 3)x - 12\theta = 0 \]
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = \theta\), \(b = -2(\theta^2 - 3)\), and \(c = -12\theta\).

Calculating the discriminant:
\[ D = [2(\theta^2 - 3)]^2 - 4\theta(-12\theta) = 4(\theta^2 - 3)^2 + 48\theta^2 \]

### Step 5: Set Conditions for the Roots

For \(f(x)\) to be defined for all \(x\) in the domain of \(g\), the roots must be outside the intervals \((-7, 7)\). Thus, we need to ensure:
1. Both roots are less than -7 or greater than 7.

### Step 6: Analyze the Quadratic Inequality

To ensure that \(f(x) \geq 0\) for \(x \in (-\infty, -7) \cup (7, \infty)\), we need to analyze the conditions on \(\theta\).

1. **Condition for Roots**:
The quadratic must be positive at \(x = 7\) and \(x = -7\):
- For \(x = 7\):
\[ \theta(7^2) - 2(\theta^2 - 3)(7) - 12\theta \geq 0 \]
- For \(x = -7\):
\[ \theta(-7^2) - 2(\theta^2 - 3)(-7) - 12\theta \geq 0 \]

### Step 7: Solve the Inequalities

By solving these inequalities, we can find the range of \(\theta\) that satisfies both conditions.

### Step 8: Combine Conditions

Finally, we combine the conditions from the well-behaved and intelligent criteria to find the complete set of values for \(\theta\).

### Final Answer

The complete set of values of \(\theta\) which are well behaved as well as intelligent is:
\[ \theta \in \left[\frac{6}{7}, \frac{\pi}{2}\right] \]