If f(x) be such that `f(x) = max (|3 - x|, 3 - x^(3))`, then

Options

A. (a) f(x) is continuous `AA x in R`
B. (b) f(x) is differentiable `AA x in R`
C. (c) f(x) is non-differentiable at three points only
D. (d) f(x) is non-differentiable at four points only

Answer

To solve the problem, we need to analyze the function \( f(x) = \max (|3 - x|, 3 - x^3) \) to determine its continuity and differentiability.

### Step 1: Understand the Components of the Function
The function consists of two parts:
1. \( |3 - x| \)
2. \( 3 - x^3 \)

### Step 2: Analyze \( |3 - x| \)
The function \( |3 - x| \) can be expressed as:
- \( 3 - x \) when \( x \leq 3 \)
- \( x - 3 \) when \( x > 3 \)

### Step 3: Analyze \( 3 - x^3 \)
The function \( 3 - x^3 \) is a cubic function that decreases as \( x \) increases. It intersects the y-axis at \( y = 3 \) when \( x = 0 \).

### Step 4: Find Intersection Points
To find where the two functions are equal, we set:
\[ |3 - x| = 3 - x^3 \]

**Case 1:** When \( x \leq 3 \):
\[ 3 - x = 3 - x^3 \]
This simplifies to:
\[ x^3 = x \]
Factoring gives:
\[ x(x^2 - 1) = 0 \]
Thus, \( x = 0, 1, -1 \).

**Case 2:** When \( x > 3 \):
\[ x - 3 = 3 - x^3 \]
This simplifies to:
\[ x^3 + x - 6 = 0 \]
Finding roots for this cubic equation may require numerical methods or graphing.

### Step 5: Determine Points of Non-Differentiability
The function \( f(x) \) will be non-differentiable at points where the two functions intersect, as well as at points where the absolute value function changes its form.

From our analysis:
- The points of intersection include \( x = 0, 1, -1 \).
- We also need to check the point \( x = 3 \) where \( |3 - x| \) changes its form.

### Step 6: Continuity of \( f(x) \)
To check continuity, we need to ensure that at all points where the function changes from one piece to another, the left-hand limit equals the right-hand limit and equals the function value at that point.

Since both components are continuous functions, and since we are taking the maximum, \( f(x) \) is continuous everywhere.

### Step 7: Count Non-Differentiable Points
From our analysis, we found:
- Non-differentiable at \( x = 0, 1, -1, 3 \).

Thus, \( f(x) \) is non-differentiable at four points.

### Final Answer
- **Continuity:** \( f(x) \) is continuous everywhere.
- **Non-differentiable Points:** \( f(x) \) is non-differentiable at 4 points.