let `f: RrarrR` be a function defined by `f(x)=max{x,x^3}`. The set of values where f(x) is differentiable is:

Options

A. {-1, 1}
B. {-1, 0}
C. {0, 1}
D. {-1, 0, 1}

Answer

To determine the set of values where the function \( f(x) = \max\{x, x^3\} \) is differentiable, we will follow these steps:

### Step 1: Understand the Function
The function \( f(x) \) takes the maximum of two functions: \( x \) and \( x^3 \). This means that for each \( x \), \( f(x) \) will be equal to either \( x \) or \( x^3 \), depending on which is larger.

### Step 2: Find Points of Intersection
To find where the two functions intersect, we set \( x = x^3 \):
\[ x^3 - x = 0 \]
Factoring gives:
\[ x(x^2 - 1) = 0 \]
This can be factored further:
\[ x(x - 1)(x + 1) = 0 \]
Thus, the points of intersection are:
\[ x = -1, \quad x = 0, \quad x = 1 \]

### Step 3: Analyze the Function in Intervals
We will analyze the function in the intervals determined by the points of intersection: \( (-\infty, -1) \), \( (-1, 0) \), \( (0, 1) \), and \( (1, \infty) \).

1. **For \( x < -1 \)**:
Here, \( x^3 < x \), so \( f(x) = x \).

2. **For \( -1 < x < 0 \)**:
Here, \( x^3 < x \), so \( f(x) = x \).

3. **For \( 0 < x < 1 \)**:
Here, \( x^3 > x \), so \( f(x) = x^3 \).

4. **For \( x > 1 \)**:
Here, \( x^3 > x \), so \( f(x) = x^3 \).

### Step 4: Determine Differentiability
The function \( f(x) \) is differentiable in each of the intervals we analyzed, except at the points where the definition of \( f(x) \) changes, which are the points of intersection \( x = -1, 0, 1 \).

To check differentiability at these points, we need to examine the left-hand and right-hand derivatives:

- At \( x = -1 \):
- Left-hand derivative: \( f'(x) = 1 \) (since \( f(x) = x \))
- Right-hand derivative: \( f'(x) = 1 \) (since \( f(x) = x \))
- Since both derivatives are equal, \( f \) is differentiable at \( x = -1 \).

- At \( x = 0 \):
- Left-hand derivative: \( f'(x) = 1 \) (since \( f(x) = x \))
- Right-hand derivative: \( f'(x) = 0 \) (since \( f(x) = x^3 \))
- Since the derivatives are not equal, \( f \) is not differentiable at \( x = 0 \).

- At \( x = 1 \):
- Left-hand derivative: \( f'(x) = 3 \) (since \( f(x) = x^3 \))
- Right-hand derivative: \( f'(x) = 3 \) (since \( f(x) = x^3 \))
- Since both derivatives are equal, \( f \) is differentiable at \( x = 1 \).

### Conclusion
The function \( f(x) \) is differentiable for all \( x \in \mathbb{R} \) except at \( x = 0 \). Thus, the set of values where \( f(x) \) is differentiable is:
\[ \text{Differentiable at } \mathbb{R} \setminus \{0\} \]