let A be the greatest value of the function `f(x)=log _(x) [x],` (where `[.]` denotes gratest integer function) and B be the least value of the function `g (x) = |sin x | +|cos x| ,` then :

Options

A. `A gtB`
B. `A lt B`
C. `A =B`
D. `2A+ B=4`

Answer

To solve the problem, we need to find the values of A and B based on the given functions and then compare them.

### Step 1: Determine A from the function \( f(x) = \log_{\lfloor x \rfloor} x \)

1. **Understanding the function**: The function \( f(x) \) involves the logarithm with the base being the greatest integer function \( \lfloor x \rfloor \).
2. **Domain**: The function is defined for \( x > 1 \) because the base of the logarithm must be greater than 0 and not equal to 1.
3. **Behavior of \( f(x) \)**:
- For \( x \) in the interval \( [n, n+1) \) where \( n \) is a positive integer, \( \lfloor x \rfloor = n \).
- Thus, \( f(x) = \log_n x \).
4. **Finding the maximum value**:
- The maximum value of \( \log_n x \) occurs when \( x = n \), giving \( f(n) = \log_n n = 1 \).
- As \( x \) approaches \( n+1 \), \( f(x) \) approaches \( \log_n(n+1) \), which is less than 1.
5. **Conclusion for A**: The greatest value of \( f(x) \) is \( A = 1 \).

### Step 2: Determine B from the function \( g(x) = |\sin x| + |\cos x| \)

1. **Understanding the function**: The function \( g(x) \) is the sum of the absolute values of sine and cosine.
2. **Finding the minimum value**:
- The minimum value of \( |\sin x| + |\cos x| \) occurs when both sine and cosine are equal. This happens at \( x = \frac{\pi}{4} + k\pi \) for any integer \( k \).
- At \( x = \frac{\pi}{4} \), \( |\sin(\frac{\pi}{4})| = |\cos(\frac{\pi}{4})| = \frac{1}{\sqrt{2}} \).
- Therefore, \( g\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2} \).
3. **Finding the maximum value**:
- The maximum value occurs when either sine or cosine is at its peak, which is \( \sqrt{2} \) when both are at their maximum.
4. **Conclusion for B**: The least value of \( g(x) \) is \( B = 1 \).

### Step 3: Compare A and B

- We have found \( A = 1 \) and \( B = 1 \).
- Therefore, \( A = B \).

### Final Answer

The relationship between A and B is:
\[ A = B \]