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 structure
The function \( f(x) \) is composed of nested absolute values. We can break it down into parts:
1. \( g(x) = x^2 - 3 \)
2. \( h(x) = |g(x)| = |x^2 - 3| \)
3. \( f(x) = |h(x) - 2| = ||x^2 - 3| - 2| \)

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

### Step 3: Analyze the function \( h(x) \)
The function \( h(x) = |x^2 - 3| \) will be:
- \( h(x) = x^2 - 3 \) for \( x^2 \geq 3 \) (i.e., \( x \leq -\sqrt{3} \) or \( x \geq \sqrt{3} \))
- \( h(x) = -(x^2 - 3) = 3 - x^2 \) for \( -\sqrt{3} < x < \sqrt{3} \)

### Step 4: Analyze the function \( f(x) \)
Now, we analyze \( f(x) = |h(x) - 2| \):
1. For \( x \leq -\sqrt{3} \) or \( x \geq \sqrt{3} \):
- \( f(x) = |(x^2 - 3) - 2| = |x^2 - 5| \)
2. For \( -\sqrt{3} < x < \sqrt{3} \):
- \( f(x) = |(3 - x^2) - 2| = |1 - x^2| \)

### Step 5: Determine the critical points
- For \( f(x) = |x^2 - 5| \):
- It equals 0 at \( x = \pm\sqrt{5} \).
- It equals 2 at \( x^2 - 5 = 2 \) or \( x^2 = 7 \) (i.e., \( x = \pm\sqrt{7} \)).

- For \( f(x) = |1 - x^2| \):
- It equals 0 at \( x = \pm 1 \).
- It equals 2 at \( 1 - x^2 = 2 \) or \( x^2 = -1 \) (not possible).

### Step 6: Sketch the graph of \( f(x) \)
- The graph of \( f(x) \) will have a V-shape at the points where it equals 0 and will change slopes at \( x = \pm\sqrt{5} \) and \( x = \pm\sqrt{3} \).

### Step 7: Determine conditions for 2 solutions
To have exactly 2 solutions for \( f(x) = \lambda \):
- The horizontal line \( y = \lambda \) must intersect the graph of \( f(x) \) at exactly two points.
- From the analysis, we find that:
- If \( \lambda < 2 \), the line intersects at 4 points.
- If \( \lambda = 2 \), the line intersects at 4 points.
- If \( \lambda > 2 \), the line intersects at 2 points.

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