Number of points where the function f(x)=`"Maximum [sgn (x)",-sqrt(9-x^(2)),x^(3)]` is continuous but not differentiable, is

Options

A. 4
B. 2
C. 5
D. 6

Explanation

To determine the number of points where the function \( f(x) = \max[\text{sgn}(x), -\sqrt{9 - x^2}, x^3] \) is continuous but not differentiable, we will analyze each component of the function and their interactions.

### Step 1: Understand the Components of the Function
1. **sgn(x)**:
- Defined as:
- \( \text{sgn}(x) = -1 \) for \( x < 0 \)
- \( \text{sgn}(x) = 0 \) for \( x = 0 \)
- \( \text{sgn}(x) = 1 \) for \( x > 0 \)

2. **-\(\sqrt{9 - x^2}\)**:
- This represents the lower semicircle with radius 3 centered at the origin. It is defined for \( -3 \leq x \leq 3 \).

3. **\(x^3\)**:
- This is a cubic function defined for all real \(x\).

### Step 2: Identify the Points of Intersection
We need to find the points where these functions intersect, as these points will be candidates for where \( f(x) \) could change its maximum value.

1. **Intersection of \( \text{sgn}(x) \) and \( -\sqrt{9 - x^2} \)**:
- For \( x < 0 \): \( -1 = -\sqrt{9 - x^2} \) leads to \( \sqrt{9 - x^2} = 1 \) or \( 9 - x^2 = 1 \) → \( x^2 = 8 \) → \( x = -\sqrt{8} \) or \( x = -2\sqrt{2} \).
- For \( x = 0 \): \( 0 = -\sqrt{9 - 0^2} \) → \( 0 = -3 \) (not valid).
- For \( x > 0 \): \( 1 = -\sqrt{9 - x^2} \) (not valid).

2. **Intersection of \( -\sqrt{9 - x^2} \) and \( x^3 \)**:
- Solve \( -\sqrt{9 - x^2} = x^3 \).
- Squaring both sides gives \( 9 - x^2 = x^6 \) → \( x^6 + x^2 - 9 = 0 \).
- Let \( y = x^2 \) → \( y^3 + y - 9 = 0 \). This cubic equation can be solved to find real roots.

3. **Intersection of \( \text{sgn}(x) \) and \( x^3 \)**:
- For \( x < 0 \): \( -1 = x^3 \) → \( x = -1 \).
- For \( x = 0 \): \( 0 = 0 \).
- For \( x > 0 \): \( 1 = x^3 \) → \( x = 1 \).

### Step 3: Analyze Continuity and Differentiability
- The function \( f(x) \) is continuous at all points where it is defined as it is the maximum of continuous functions.
- However, it is not differentiable at points where the maximum switches from one function to another, particularly at sharp edges.

### Step 4: Identify Points of Non-Differentiability
From our analysis:
1. At \( x = -2\sqrt{2} \) (intersection of \( \text{sgn}(x) \) and \( -\sqrt{9 - x^2} \)).
2. At \( x = -1 \) (intersection of \( \text{sgn}(x) \) and \( x^3 \)).
3. At \( x = 0 \) (intersection of all three functions).
4. At \( x = 1 \) (intersection of \( x^3 \) and \( \text{sgn}(x) \)).
5. At points where \( -\sqrt{9 - x^2} \) intersects \( x^3 \) (roots of \( y^3 + y - 9 = 0 \)).

### Conclusion
After analyzing the intersections and points of non-differentiability, we find that there are **4 points** where the function is continuous but not differentiable.

### Final Answer
The number of points where the function \( f(x) \) is continuous but not differentiable is **4**.