The function `f: (a, oo) ->R` where R denotes the range corresponding to the given domain, with rule `f(x)=2x^3-3x^2 +6`, will have an inverse provided

Options

A. `a le1`
B. `age0`
C. `ale0`
D. `age1`

Explanation

To determine if the function \( f(x) = 2x^3 - 3x^2 + 6 \) has an inverse, we need to check if it is a one-to-one function (injective) and onto (surjective) over the given domain \( (a, \infty) \).

### Step 1: Find the derivative of the function.
To check for monotonicity, we first differentiate the function:

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

### Step 2: Set the derivative to zero to find critical points.
Next, we set the derivative equal to zero to find critical points:

\[ 6x^2 - 6 = 0 \]

Dividing by 6 gives:

\[ x^2 - 1 = 0 \]

Factoring gives:

\[ (x - 1)(x + 1) = 0 \]

Thus, the critical points are:

\[ x = 1 \quad \text{and} \quad x = -1 \]

### Step 3: Analyze the sign of the derivative.
To determine the intervals of increase and decrease, we can test the sign of \( f'(x) \) in the intervals defined by the critical points:

1. For \( x < -1 \): Choose \( x = -2 \):
\[ f'(-2) = 6(-2)^2 - 6 = 24 - 6 = 18 \quad (\text{positive, increasing}) \]

2. For \( -1 < x < 1 \): Choose \( x = 0 \):
\[ f'(0) = 6(0)^2 - 6 = -6 \quad (\text{negative, decreasing}) \]

3. For \( x > 1 \): Choose \( x = 2 \):
\[ f'(2) = 6(2)^2 - 6 = 24 - 6 = 18 \quad (\text{positive, increasing}) \]

### Step 4: Determine the behavior of the function.
From the analysis, we see that:

- The function is increasing on \( (-\infty, -1) \).
- The function is decreasing on \( (-1, 1) \).
- The function is increasing on \( (1, \infty) \).

### Step 5: Identify the minimum value.
The minimum value occurs at \( x = 1 \). We can find \( f(1) \):

\[ f(1) = 2(1)^3 - 3(1)^2 + 6 = 2 - 3 + 6 = 5 \]

### Step 6: Conclusion about the inverse.
For the function to have an inverse, it must be one-to-one. Since the function is decreasing between \( (-1, 1) \) and increasing afterwards, it is not one-to-one over the entire domain. However, if we restrict the domain to \( [1, \infty) \), the function becomes one-to-one.

Thus, the function \( f(x) \) will have an inverse provided \( a \geq 1 \).

### Final Answer:
The function \( f(x) = 2x^3 - 3x^2 + 6 \) will have an inverse provided \( a \geq 1 \).