The function `f(x) = 4 -3x + 3x^(2) - x^(3)` is

Options

A. decreasing on R
B. increasing on R
C. strictly decreasing on R
D. strictly increasing on R

Explanation

To determine whether the function \( f(x) = 4 - 3x + 3x^2 - x^3 \) is increasing or decreasing, we need to find its derivative and analyze the sign of the derivative.

### Step-by-Step Solution:

1. **Find the Derivative**:
We start by differentiating the function \( f(x) \).
\[ f'(x) = \frac{d}{dx}(4 - 3x + 3x^2 - x^3) \]
Using the power rule, we differentiate each term:
\[ f'(x) = 0 - 3 + 6x - 3x^2 \]
Thus, we can simplify this to:
\[ f'(x) = -3 + 6x - 3x^2 \]

2. **Rearranging the Derivative**:
We can rearrange \( f'(x) \):
\[ f'(x) = -3x^2 + 6x - 3 \]
Factoring out \(-3\):
\[ f'(x) = -3(x^2 - 2x + 1) \]

3. **Factoring the Quadratic**:
The quadratic \( x^2 - 2x + 1 \) can be factored as:
\[ f'(x) = -3(x - 1)^2 \]

4. **Analyzing the Sign of the Derivative**:
The expression \( (x - 1)^2 \) is always non-negative (i.e., \( (x - 1)^2 \geq 0 \)) for all \( x \). Therefore, \( -3(x - 1)^2 \) is always less than or equal to zero:
\[ f'(x) \leq 0 \]
This indicates that the function \( f(x) \) is non-increasing for all \( x \).

5. **Conclusion**:
Since the derivative \( f'(x) \) is less than or equal to zero for all \( x \), we conclude that the function \( f(x) \) is a decreasing function on its entire domain.