If the function of `f(x) = [((x-5)^(2))/(A)]sin(x-5)+a cos(x - 2),"where"[*]` denotes the greatest integer function, is continuous and differentiable in (7, 9), then find the value of `A`

Options

A. `A in [8, 64]`
B. `A in [0, 8)`
C. `A in [16, oo)`
D. `A in [8, 16]`

Answer

To solve the problem, we need to find the value of \( A \) such that the function

\[ f(x) = \left\lfloor \frac{(x-5)^2}{A} \right\rfloor \sin(x-5) + A \cos(x-2) \]

is continuous and differentiable in the interval \( (7, 9) \). Here, \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

### Step 1: Analyze the function in the given interval

We first note that \( (x-5)^2 \) will vary as \( x \) goes from 7 to 9:

- When \( x = 7 \):
\[ (7-5)^2 = 2^2 = 4 \]
- When \( x = 9 \):
\[ (9-5)^2 = 4^2 = 16 \]

Thus, as \( x \) varies from 7 to 9, \( (x-5)^2 \) varies from 4 to 16.

### Step 2: Determine the range of the greatest integer function

The greatest integer function \( \left\lfloor \frac{(x-5)^2}{A} \right\rfloor \) will take integer values depending on the value of \( A \):

- For \( A > 16 \):
\[ \frac{(x-5)^2}{A} \text{ will be between } \frac{4}{A} \text{ and } \frac{16}{A} \]
This means \( \left\lfloor \frac{(x-5)^2}{A} \right\rfloor = 0 \) for \( A > 16 \).

### Step 3: Continuity condition

For \( f(x) \) to be continuous, we need:

\[ \left\lfloor \frac{(x-5)^2}{A} \right\rfloor \sin(x-5) + A \cos(x-2) \]

to be continuous. Since \( \left\lfloor \frac{(x-5)^2}{A} \right\rfloor = 0 \) when \( A > 16 \), we can simplify \( f(x) \):

\[ f(x) = A \cos(x-2) \]

This function is continuous and differentiable for all \( x \).

### Step 4: Differentiability condition

Since \( A \cos(x-2) \) is differentiable for all \( x \), we do not have any issues with differentiability as long as \( A > 16 \).

### Conclusion

Thus, for the function \( f(x) \) to be continuous and differentiable in the interval \( (7, 9) \), we require:

\[ A > 16 \]

Therefore, the value of \( A \) must be greater than 16.

### Final Answer
The value of \( A \) is \( A > 16 \).