Let `f(x) = x^(3) - 6x^(2) + 15x + 3`. Then

Options

A. `f(X) gt 0` for all `x in R`
B. `f(x) gt f(x+1`) does not hold for any real x.
C. f(x) is invertible.
D. f(x) is a one - one function

Answer

To solve the problem, we need to analyze the function \( f(x) = x^3 - 6x^2 + 15x + 3 \). We will determine the nature of the function by finding its derivative and analyzing its behavior.

### Step 1: Find the derivative of \( f(x) \)

The first step is to differentiate the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(x^3 - 6x^2 + 15x + 3) \]

Using the power rule of differentiation:

\[ f'(x) = 3x^2 - 12x + 15 \]

### Step 2: Analyze the derivative

Next, we need to analyze the derivative \( f'(x) \) to determine where it is positive or negative. This will help us understand the behavior of the function \( f(x) \).

To do this, we can find the discriminant of the quadratic equation \( 3x^2 - 12x + 15 \).

The discriminant \( D \) is given by:

\[ D = b^2 - 4ac \]

where \( a = 3 \), \( b = -12 \), and \( c = 15 \).

Calculating the discriminant:

\[ D = (-12)^2 - 4 \cdot 3 \cdot 15 = 144 - 180 = -36 \]

Since the discriminant is less than 0, the quadratic has no real roots and opens upwards (as the coefficient of \( x^2 \) is positive). This means that \( f'(x) > 0 \) for all \( x \).

### Step 3: Conclusion about the function

Since \( f'(x) > 0 \) for all \( x \), the function \( f(x) \) is strictly increasing.

### Step 4: Determine if the function is one-to-one and onto

1. **One-to-One**: Since \( f(x) \) is strictly increasing, it is a one-to-one function.
2. **Onto**: As a polynomial function of degree 3, the range of \( f(x) \) is all real numbers \( \mathbb{R} \).

Thus, \( f(x) \) is a bijective function (both one-to-one and onto).

### Step 5: Check the options

From the analysis, we can conclude:
- The function is increasing, hence it is one-to-one.
- The function is a polynomial of odd degree, so it covers all real numbers, making it onto.

### Final Answer

Thus, the correct options based on the analysis are:
- \( f(x) \) is a one-to-one function.
- \( f(x) \) is a bijective function.