A function f(x) having the following properties,
(i) f(x) is continuous except at x=3
(ii) f(x) is differentiable except at x=-2 and x=3
(iii) `f(0) =0 lim_(x to 3) f(x) to - oo lim_(x to oo) f(x) =3 , lim_(x to oo) f(x)=0`
(iv) `f'(x) gt 0 AA in (-oo, -2) uu (3,oo) " and " f'(x) le 0 AA x in (-2,3)`
(v) `f''(x) gt 0 AA x in (-oo,-2) uu (-2,0)" and "f''(x) lt 0 AA x in (0,3) uu(3,oo)`
Then answer the following questions
Find the Maximum possible number of solutions of `f(x)=|x|`

Answer

To find the maximum possible number of solutions for the equation \( f(x) = |x| \), we will analyze the properties of the function \( f(x) \) step by step.

### Step 1: Analyzing the Properties of \( f(x) \)

1. **Continuity**: The function \( f(x) \) is continuous except at \( x = 3 \).
2. **Differentiability**: The function \( f(x) \) is differentiable except at \( x = -2 \) and \( x = 3 \).
3. **Function Values**:
- \( f(0) = 0 \)
- \( \lim_{x \to 3} f(x) = -\infty \)
- \( \lim_{x \to -\infty} f(x) = 3 \)
- \( \lim_{x \to \infty} f(x) = 0 \)

4. **First Derivative**:
- \( f'(x) > 0 \) in the intervals \( (-\infty, -2) \) and \( (3, \infty) \) (function is increasing).
- \( f'(x) \leq 0 \) in the interval \( (-2, 3) \) (function is decreasing).

5. **Second Derivative**:
- \( f''(x) > 0 \) in the intervals \( (-\infty, -2) \) and \( (-2, 0) \) (function is concave up).
- \( f''(x) < 0 \) in the intervals \( (0, 3) \) and \( (3, \infty) \) (function is concave down).

### Step 2: Sketching the Graph of \( f(x) \)

Based on the properties above, we can sketch the graph of \( f(x) \):

- From \( -\infty \) to \( -2 \), \( f(x) \) is increasing and concave up.
- At \( x = -2 \), the function has a corner (not differentiable but continuous).
- From \( -2 \) to \( 0 \), \( f(x) \) is decreasing and concave down.
- At \( x = 0 \), \( f(0) = 0 \).
- From \( 0 \) to \( 3 \), \( f(x) \) continues to decrease and approaches \( -\infty \) as \( x \) approaches \( 3 \).
- At \( x = 3 \), there is a discontinuity (not continuous).
- From \( 3 \) to \( \infty \), \( f(x) \) is increasing and approaches \( 0 \).

### Step 3: Sketching the Graph of \( |x| \)

The graph of \( |x| \) is a V-shaped graph that intersects the x-axis at the origin and has a slope of 1 for \( x \geq 0 \) and a slope of -1 for \( x < 0 \).

### Step 4: Finding Intersections

To find the maximum number of solutions to \( f(x) = |x| \):

1. **From \( -\infty \) to \( -2 \)**: The function is increasing. It can intersect \( |x| \) once in this interval.
2. **At \( x = -2 \)**: The function has a corner, which can lead to another intersection.
3. **From \( -2 \) to \( 0 \)**: The function is decreasing. It can intersect \( |x| \) once more.
4. **At \( x = 0 \)**: The function intersects \( |x| \) at the origin.
5. **From \( 0 \) to \( 3 \)**: The function is decreasing and approaches \( -\infty \). It can intersect \( |x| \) once more.
6. **From \( 3 \) to \( \infty \)**: The function is increasing and approaches \( 0 \). It can intersect \( |x| \) once more.

### Conclusion

Counting all possible intersections, we find:
- 1 intersection in \( (-\infty, -2) \)
- 1 intersection at \( x = -2 \)
- 1 intersection in \( (-2, 0) \)
- 1 intersection at \( x = 0 \)
- 1 intersection in \( (0, 3) \)
- 1 intersection in \( (3, \infty) \)

Thus, the maximum possible number of solutions for \( f(x) = |x| \) is **5**.