. For `x in R^+, if x, [x], {x}` are in harmonic progression then the value of x can not be equal to (where [*] denotes greatest integer function, {*} denotes fractional part function)

Options

A. `(1)/(sqrt2) tan ""pi/8`
B. ` (1)/(sqrt2) cot ""pi/8`
C. `(1)/(sqrt2) tan ""(pi)/(12)`
D. `(1)/(sqrt2) cot ""(pi)/(12)`

Answer

To solve the problem, we need to determine the values of \( x \) such that \( x \), \( [x] \) (the greatest integer function), and \( \{x\} \) (the fractional part function) are in harmonic progression.

### Step-by-Step Solution:

1. **Understanding Harmonic Progression**:
For three numbers \( a \), \( b \), and \( c \) to be in harmonic progression, the following condition must hold:
\[ \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \]
In our case, let \( a = x \), \( b = [x] \), and \( c = \{x\} \).

2. **Expressing the Fractional Part**:
The fractional part \( \{x\} \) can be expressed as:
\[ \{x\} = x - [x] \]
Therefore, we can rewrite the harmonic progression condition as:
\[ \frac{2}{[x]} = \frac{1}{x} + \frac{1}{x - [x]} \]

3. **Finding a Common Denominator**:
The right-hand side can be combined using a common denominator:
\[ \frac{1}{x} + \frac{1}{x - [x]} = \frac{(x - [x]) + x}{x(x - [x])} = \frac{2x - [x]}{x(x - [x])} \]
Thus, we have:
\[ \frac{2}{[x]} = \frac{2x - [x]}{x(x - [x])} \]

4. **Cross Multiplying**:
Cross-multiplying gives us:
\[ 2x(x - [x]) = [x](2x - [x]) \]

5. **Expanding Both Sides**:
Expanding both sides results in:
\[ 2x^2 - 2x[x] = 2x[x] - [x]^2 \]

6. **Rearranging the Equation**:
Rearranging the terms gives us:
\[ [x]^2 - 4x[x] + 2x^2 = 0 \]

7. **Identifying as a Quadratic Equation**:
This is a quadratic equation in terms of \( [x] \):
\[ [x]^2 - 4x[x] + 2x^2 = 0 \]

8. **Using the Quadratic Formula**:
We can apply the quadratic formula:
\[ [x] = \frac{4x \pm \sqrt{(4x)^2 - 4 \cdot 1 \cdot 2x^2}}{2 \cdot 1} \]
Simplifying this gives:
\[ [x] = \frac{4x \pm \sqrt{16x^2 - 8x^2}}{2} = \frac{4x \pm \sqrt{8x^2}}{2} = \frac{4x \pm 2\sqrt{2}x}{2} = 2x \pm \sqrt{2}x \]

9. **Finding Possible Values for [x]**:
This leads to two possible values for \( [x] \):
\[ [x] = (2 + \sqrt{2})x \quad \text{or} \quad [x] = (2 - \sqrt{2})x \]

10. **Determining Validity**:
Since \( [x] \) must be an integer, we need to check if these expressions yield integer values for \( x \).

11. **Checking Given Options**:
We need to check which of the given options cannot be equal to \( x \) based on the integer condition derived from \( [x] \).

### Conclusion:
After checking the values, we find that \( x \) cannot be equal to certain values based on the conditions derived from the harmonic progression.