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

Options

A. `lambda gt 2`
B. `lambda lt 2`
C. `1 lt lambda lt 2`
D. `lambda ge 2`

Answer

To solve the problem, we need to analyze the function \( f(x) = ||x^2 - 3| - 2| \) and determine the conditions under which the equation \( f(x) = \lambda \) has exactly 2 solutions.

### Step 1: Understand the Function
The function \( f(x) \) consists of nested absolute values. We can break it down step by step.

1. Start with the inner function \( g(x) = x^2 - 3 \).
2. The absolute value \( |g(x)| = |x^2 - 3| \) will have a minimum value of 0 when \( x^2 = 3 \), i.e., at \( x = \pm \sqrt{3} \).

### Step 2: Graph the Inner Function
The graph of \( g(x) = x^2 - 3 \) is a parabola opening upwards with its vertex at (0, -3). The points where it intersects the x-axis are \( x = -\sqrt{3} \) and \( x = \sqrt{3} \).

### Step 3: Graph the Absolute Value
The graph of \( |g(x)| = |x^2 - 3| \) reflects the part of the parabola that is below the x-axis above the x-axis. Thus, the graph will touch the x-axis at \( x = -\sqrt{3} \) and \( x = \sqrt{3} \) and rise upwards from there.

### Step 4: Shift Down by 2
Next, we consider \( |g(x)| - 2 \):
- The graph of \( |g(x)| \) is shifted down by 2 units. The new minimum value will be at \( y = -2 \) when \( x = -\sqrt{3} \) and \( x = \sqrt{3} \).

### Step 5: Final Absolute Value
Now we consider the function \( f(x) = ||g(x)| - 2| \). This means we take the absolute value of the entire expression, which will reflect any part of the graph that is below the x-axis back above it.

### Step 6: Analyze the Graph for Solutions
To find when \( f(x) = \lambda \) has exactly 2 solutions, we need to analyze the horizontal line \( y = \lambda \):
- If \( \lambda < -2 \): The line does not intersect the graph at all (0 solutions).
- If \( \lambda = -2 \): The line touches the graph at two points (1 solution).
- If \( -2 < \lambda < 0 \): The line intersects the graph at four points (4 solutions).
- If \( \lambda = 0 \): The line intersects at four points (4 solutions).
- If \( 0 < \lambda < 1 \): The line intersects at six points (6 solutions).
- If \( \lambda = 1 \): The line intersects at six points (6 solutions).
- If \( 1 < \lambda < 2 \): The line intersects at four points (4 solutions).
- If \( \lambda = 2 \): The line intersects at four points (4 solutions).
- If \( \lambda > 2 \): The line intersects the graph at exactly two points (2 solutions).

### Conclusion
Thus, the equation \( f(x) = \lambda \) has exactly 2 solutions if:
\[ \lambda > 2 \]