Let a function `f (x)=` be such that `f(x)=||x^(2)-3|-2|`.
Equations `f (x)=lambda` has 8 solutions if :

Options

A. `lambda lt 1`
B. `lambda =2`
C. `1 lt lambda lt 2`
D. `0 lt lambda lt 1`

Answer

To solve the problem step by step, we need to analyze the function \( f(x) = ||x^2 - 3| - 2| \) and find the values of \( \lambda \) for which the equation \( f(x) = \lambda \) has 8 solutions.

### Step 1: Analyze the inner function \( g(x) = x^2 - 3 \)
The function \( g(x) = x^2 - 3 \) is a parabola that opens upwards. Its vertex is at \( (0, -3) \) and it intersects the x-axis at \( x = \pm \sqrt{3} \).

**Hint:** Identify the vertex and x-intercepts of the parabola.

### Step 2: Graph the function \( |g(x)| = |x^2 - 3| \)
The graph of \( |g(x)| \) will reflect the portion of \( g(x) \) that is below the x-axis (i.e., where \( g(x) < 0 \)) across the x-axis. This means:
- For \( x < -\sqrt{3} \) and \( x > \sqrt{3} \), \( |g(x)| = g(x) \).
- For \( -\sqrt{3} < x < \sqrt{3} \), \( |g(x)| = -g(x) = 3 - x^2 \).

**Hint:** Sketch the graph of \( |g(x)| \) based on the behavior of \( g(x) \).

### Step 3: Analyze the function \( h(x) = |g(x)| - 2 \)
Now, we will shift the graph of \( |g(x)| \) down by 2 units to get \( h(x) = |g(x)| - 2 \). This will affect the y-values of the graph:
- The points where \( |g(x)| = 2 \) will be the new x-intercepts of \( h(x) \).

**Hint:** Determine where \( |g(x)| = 2 \) to find the x-intercepts of \( h(x) \).

### Step 4: Solve \( |g(x)| = 2 \)
We need to solve the equations:
1. \( x^2 - 3 = 2 \) leading to \( x^2 = 5 \) or \( x = \pm \sqrt{5} \).
2. \( x^2 - 3 = -2 \) leading to \( x^2 = 1 \) or \( x = \pm 1 \).

Thus, the solutions for \( |g(x)| = 2 \) are \( x = -\sqrt{5}, -1, 1, \sqrt{5} \).

**Hint:** Identify the points where \( |g(x)| = 2 \) to find the x-intercepts.

### Step 5: Analyze the function \( f(x) = |h(x)| \)
Now, we need to take the absolute value of \( h(x) \). The graph of \( f(x) = ||g(x)| - 2| \) will reflect any part of \( h(x) \) that is below the x-axis across the x-axis.

**Hint:** Understand how the absolute value affects the graph of \( h(x) \).

### Step 6: Determine the number of solutions for \( f(x) = \lambda \)
To find when \( f(x) = \lambda \) has 8 solutions, we need to consider horizontal lines \( y = \lambda \):
- For \( \lambda < 0 \): No intersections.
- For \( \lambda = 0 \): 4 intersections (from the points found earlier).
- For \( 0 < \lambda < 1 \): The horizontal line will intersect the graph 8 times.
- For \( \lambda = 1 \): 7 intersections.
- For \( \lambda \geq 1 \): Fewer than 8 intersections.

**Hint:** Count the intersections for different values of \( \lambda \).

### Conclusion
The values of \( \lambda \) for which the equation \( f(x) = \lambda \) has 8 solutions are in the interval \( (0, 1) \).

**Final Answer:** \( \lambda \) is in the range \( (0, 1) \).