Let `f(x) = x^(2) - 5x + 6, g(x) = f(|x|), h(x) = |g(x)|`
The set of value of `mu` such that equation `h(x) = mu` has exactly `8` real and distinct roots, contains. (a) 0 (b) `1/8` (c) `1/16` (d) `1/4`

Options

A. `0`
B. `(1)/(8)`
C. `(1)/(16)`
D. `(1)/(4)`

Explanation

To solve the problem step by step, we will analyze the functions given and determine the conditions under which the equation \( h(x) = \mu \) has exactly 8 real and distinct roots.

### Step 1: Analyze the function \( f(x) \)
The function is given as:
\[ f(x) = x^2 - 5x + 6 \]
We can factor this quadratic:
\[ f(x) = (x - 2)(x - 3) \]
This means the roots of \( f(x) \) are \( x = 2 \) and \( x = 3 \).

**Hint:** Factor the quadratic to find its roots.

### Step 2: Define \( g(x) \)
Next, we define \( g(x) = f(|x|) \):
\[ g(x) = f(|x|) = (|x| - 2)(|x| - 3) \]
This function will have the same roots as \( f(x) \) but will be symmetric about the y-axis due to the absolute value.

**Hint:** Remember that \( |x| \) affects the symmetry of the function.

### Step 3: Analyze \( g(x) \)
The roots of \( g(x) \) occur at \( |x| = 2 \) and \( |x| = 3 \). Therefore, the roots of \( g(x) \) are:
- \( x = -3, -2, 2, 3 \)

**Hint:** Identify the roots of \( g(x) \) based on the symmetry of \( |x| \).

### Step 4: Define \( h(x) \)
Now we define \( h(x) = |g(x)| \):
\[ h(x) = |(|x| - 2)(|x| - 3)| \]
This function will also be symmetric and will touch the x-axis at the points where \( g(x) = 0 \).

**Hint:** The absolute value will affect the behavior of the function at the roots.

### Step 5: Analyze the behavior of \( h(x) \)
The function \( h(x) \) will have local maxima and minima. The critical points occur at \( x = -3, -2, 2, 3 \). Between these points, the function will change behavior based on the sign of \( g(x) \).

**Hint:** Sketch the graph of \( h(x) \) to visualize its behavior.

### Step 6: Determine the number of roots for \( h(x) = \mu \)
To have exactly 8 real and distinct roots for \( h(x) = \mu \), the horizontal line \( y = \mu \) must intersect the graph of \( h(x) \) at 8 distinct points. This occurs when \( \mu \) is between the minimum value of \( h(x) \) and the maximum value of \( h(x) \).

### Step 7: Find the range of \( \mu \)
From the graph of \( h(x) \), we can observe that:
- The minimum value of \( h(x) \) occurs at \( \mu = 0 \).
- The maximum value occurs at \( \mu = \frac{1}{4} \).

Thus, for \( h(x) = \mu \) to have exactly 8 real and distinct roots, \( \mu \) must be in the range:
\[ 0 < \mu < \frac{1}{4} \]

### Conclusion
The set of values of \( \mu \) such that the equation \( h(x) = \mu \) has exactly 8 real and distinct roots contains values in the interval \( (0, \frac{1}{4}) \). Therefore, the correct answer is:
- (a) 0
- (b) \( \frac{1}{8} \)
- (c) \( \frac{1}{16} \)
- (d) \( \frac{1}{4} \)

The correct options that are in the range \( (0, \frac{1}{4}) \) are \( \frac{1}{8} \) and \( \frac{1}{16} \).

**Final Answer:** The set of values of \( \mu \) such that \( h(x) = \mu \) has exactly 8 real and distinct roots contains \( \frac{1}{8} \) and \( \frac{1}{16} \).