The largest interval lying in `(-pi/2,pi/2)` for which the function `[f(x)=4^-x^2+cos^(-1)(x/2-1)+log(cosx)]` is defined, is (1) `[0,pi]` (2) `(-pi/2,pi/2)` (3) `[-pi/4,pi/2)` (4) `[0,pi/2)`

Options

A. `[0, pi]`
B. `(-(pi)/(2), (pi)/(2))`
C. `[-(pi)/(4), (pi)/(2))`
D. `[0, (pi)/(2))`

Answer

To determine the largest interval lying in \((- \frac{\pi}{2}, \frac{\pi}{2})\) for which the function

\[ f(x) = 4^{-x^2} + \cos^{-1}\left(\frac{x}{2} - 1\right) + \log(\cos x) \]

is defined, we need to analyze each component of the function.

### Step 1: Analyze \(4^{-x^2}\)

The term \(4^{-x^2}\) is defined for all real numbers \(x\). Therefore, there are no restrictions on this part of the function.

**Hint:** This term is an exponential function and is defined for all \(x\).

### Step 2: Analyze \(\cos^{-1}\left(\frac{x}{2} - 1\right)\)

The function \(\cos^{-1}(y)\) is defined for \(y\) in the interval \([-1, 1]\). Thus, we need to set up the inequality:

\[ -1 \leq \frac{x}{2} - 1 \leq 1 \]

Solving these inequalities:

1. From \(\frac{x}{2} - 1 \geq -1\):
\[ \frac{x}{2} \geq 0 \implies x \geq 0 \]

2. From \(\frac{x}{2} - 1 \leq 1\):
\[ \frac{x}{2} \leq 2 \implies x \leq 4 \]

Combining these results, we find that:

\[ 0 \leq x \leq 4 \]

**Hint:** The range of the argument of the \(\cos^{-1}\) function must be between -1 and 1.

### Step 3: Analyze \(\log(\cos x)\)

The logarithmic function \(\log(y)\) is defined for \(y > 0\). Therefore, we require:

\[ \cos x > 0 \]

The cosine function is positive in the intervals:

\[ (-\frac{\pi}{2}, \frac{\pi}{2}) \]

Within this interval, \(\cos x\) is positive.

**Hint:** Identify where the cosine function is positive to ensure the logarithm is defined.

### Step 4: Combine the results

Now we combine the results from Steps 2 and 3:

- From Step 2, we found that \(0 \leq x \leq 4\).
- From Step 3, we found that \(x\) must be in the interval \((- \frac{\pi}{2}, \frac{\pi}{2})\).

The intersection of these two intervals is:

\[ [0, \frac{\pi}{2}) \]

This is the largest interval lying in \((- \frac{\pi}{2}, \frac{\pi}{2})\) where the function \(f(x)\) is defined.

### Final Answer:

The largest interval for which the function is defined is:

\[ [0, \frac{\pi}{2}) \]

### Conclusion:

Thus, the correct option is (4) \([0, \frac{\pi}{2})\).