Let `f(x) lt 0 AA x in (-oo, 0) and f (x) gt 0 ,AA x in (0,oo) ` also `f (0)=0,` Again `f'(x) lt 0 ,AA x in (-oo, -1) and f '(x) gt 0, AA x in (-1,oo)` also `f '(-1)=0` given `lim _(x to -oo) f (x)=0 and lim _(x to oo) f (x)=oo` and function is twice differentiable.
If `f'(x) lt 0 AA x in (0,oo)and f'(0)=1` then number of solutions of equation `f (x)=x ^(2)` is : (a) 1 (b) 2 (c) 3 (d) 4

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to analyze the function \( f(x) \) based on the given conditions and determine the number of solutions to the equation \( f(x) = x^2 \).

### Step-by-Step Solution:

1. **Understanding the Function Behavior**:
- We know that \( f(x) < 0 \) for \( x \in (-\infty, 0) \) and \( f(x) > 0 \) for \( x \in (0, \infty) \).
- This implies that the function crosses the x-axis at \( x = 0 \) since \( f(0) = 0 \).

2. **Analyzing the Derivative**:
- The derivative \( f'(x) < 0 \) for \( x \in (-\infty, -1) \) indicates that \( f(x) \) is decreasing in this interval.
- The derivative \( f'(x) > 0 \) for \( x \in (-1, \infty) \) indicates that \( f(x) \) is increasing in this interval.
- Since \( f'(-1) = 0 \), we have a local minimum at \( x = -1 \).

3. **Limits of the Function**:
- Given \( \lim_{x \to -\infty} f(x) = 0 \) and \( \lim_{x \to \infty} f(x) = \infty \), we can conclude that as \( x \) approaches negative infinity, \( f(x) \) approaches 0 from below, and as \( x \) approaches positive infinity, \( f(x) \) approaches infinity.

4. **Behavior at Key Points**:
- At \( x = -1 \), since \( f(x) \) is decreasing before this point and increasing after, we can infer that \( f(-1) < 0 \) (since \( f(x) < 0 \) for \( x < 0 \)).
- At \( x = 0 \), \( f(0) = 0 \).
- For \( x > 0 \), since \( f'(x) < 0 \) implies that \( f(x) \) is decreasing, and \( f(0) = 0 \), we conclude that \( f(x) \) will remain positive for \( x > 0 \).

5. **Graphical Representation**:
- We can sketch the graph of \( f(x) \):
- It starts from \( 0 \) at \( x = 0 \), goes down to a minimum at \( x = -1 \), and then increases towards \( +\infty \) as \( x \) increases.
- The graph of \( y = x^2 \) is a parabola opening upwards, intersecting the y-axis at \( (0,0) \).

6. **Finding Intersections**:
- We need to find the intersection points of \( f(x) \) and \( x^2 \).
- Since \( f(x) < 0 \) for \( x < 0 \) and \( x^2 > 0 \) for \( x < 0 \), there are no intersections in this region.
- For \( x = 0 \), \( f(0) = 0 \) and \( x^2 = 0 \), which is one intersection.
- For \( x > 0 \), since \( f(x) \) is positive and decreasing, and \( x^2 \) is increasing, there will be two intersections: one where \( f(x) \) starts above \( x^2 \) and decreases to meet it, and another where it meets again as \( x^2 \) increases.

### Conclusion:
- Therefore, the total number of solutions to the equation \( f(x) = x^2 \) is **2**.

### Final Answer:
(b) 2