The range of values of a so that all the roots of the equations `2x^(3)-3x^(2)-12x+a=0` are real and distinct, belongs to

Options

A. (7,20)
B. (-7,20)
C. (-20,7)
D. (-7,7)

Answer

To find the range of values of \( a \) such that all the roots of the equation \( 2x^3 - 3x^2 - 12x + a = 0 \) are real and distinct, we will follow these steps:

### Step 1: Define the function
Let \( f(x) = 2x^3 - 3x^2 - 12x + a \).

### Step 2: Find the derivative
To determine the critical points where the maxima and minima occur, we need to find the derivative of the function:
\[ f'(x) = \frac{d}{dx}(2x^3 - 3x^2 - 12x + a) = 6x^2 - 6x - 12. \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ 6x^2 - 6x - 12 = 0. \]
Dividing through by 6 gives:
\[ x^2 - x - 2 = 0. \]

### Step 4: Factor the quadratic
We can factor this quadratic:
\[ (x - 2)(x + 1) = 0. \]
Thus, the critical points are:
\[ x = 2 \quad \text{and} \quad x = -1. \]

### Step 5: Evaluate the function at critical points
Now we will evaluate \( f(x) \) at these critical points to find the conditions for real and distinct roots.

1. **Evaluate \( f(-1) \)**:
\[ f(-1) = 2(-1)^3 - 3(-1)^2 - 12(-1) + a = -2 - 3 + 12 + a = 7 + a. \]
For \( f(-1) \) to be greater than zero:
\[ 7 + a > 0 \implies a > -7. \]

2. **Evaluate \( f(2) \)**:
\[ f(2) = 2(2)^3 - 3(2)^2 - 12(2) + a = 16 - 12 - 24 + a = -20 + a. \]
For \( f(2) \) to be less than zero:
\[ -20 + a < 0 \implies a < 20. \]

### Step 6: Combine the inequalities
From the evaluations, we have:
\[ -7 < a < 20. \]
Thus, the range of values of \( a \) for which all roots of the equation are real and distinct is:
\[ a \in (-7, 20). \]

### Final Answer
The range of values of \( a \) such that all the roots of the equation \( 2x^3 - 3x^2 - 12x + a = 0 \) are real and distinct is:
\[ (-7, 20). \]