Let f(x) =`x^(3)-3(7-a)x^(2)-3(9-a^(2))x+2`
The values of parameter a if f(x) has points of extrema which are opposite in sign are

Options

A. `pi`
B. (-3,3)
C. `(-oo,(58)/(14))`
D. none of these

Explanation

To solve the problem, we need to analyze the function \( f(x) = x^3 - 3(7-a)x^2 - 3(9-a^2)x + 2 \) and find the values of the parameter \( a \) such that the points of extrema of \( f(x) \) have opposite signs.

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

First, we differentiate \( f(x) \) to find \( f'(x) \):

\[ f'(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(3(7-a)x^2) - \frac{d}{dx}(3(9-a^2)x) + \frac{d}{dx}(2) \]

Calculating each term:

\[ f'(x) = 3x^2 - 6(7-a)x - 3(9-a^2) \]

Thus, we have:

\[ f'(x) = 3x^2 - 6(7-a)x - 3(9-a^2) \]

### Step 2: Set the derivative equal to zero

To find the points of extrema, we set \( f'(x) = 0 \):

\[ 3x^2 - 6(7-a)x - 3(9-a^2) = 0 \]

Dividing the entire equation by 3 simplifies it to:

\[ x^2 - 2(7-a)x - (9-a^2) = 0 \]

### Step 3: Calculate the discriminant

For the quadratic equation \( Ax^2 + Bx + C = 0 \), the discriminant \( D \) is given by \( D = B^2 - 4AC \). Here, \( A = 1 \), \( B = -2(7-a) \), and \( C = -(9-a^2) \).

Calculating the discriminant:

\[ D = [-2(7-a)]^2 - 4(1)(-(9-a^2)) \]

\[ D = 4(7-a)^2 + 4(9-a^2) \]

Factoring out the 4:

\[ D = 4[(7-a)^2 + (9-a^2)] \]

### Step 4: Set the discriminant greater than zero

For the quadratic to have real roots (extrema), we need \( D \geq 0 \):

\[ (7-a)^2 + (9-a^2) \geq 0 \]

### Step 5: Analyze the condition for opposite signs

To have extrema with opposite signs, we need to analyze the conditions on the roots of the quadratic:

1. The sum of the roots \( r_1 + r_2 = -\frac{B}{A} = 2(7-a) \) should be positive.
2. The product of the roots \( r_1 r_2 = \frac{C}{A} = -(9-a^2) \) should be negative.

#### Condition 1: Sum of roots

\[ 2(7-a) > 0 \implies 7-a > 0 \implies a < 7 \]

#### Condition 2: Product of roots

\[ -(9-a^2) < 0 \implies 9-a^2 > 0 \implies a^2 < 9 \implies -3 < a < 3 \]

### Step 6: Combine the conditions

From the conditions derived:

1. \( a < 7 \)
2. \( -3 < a < 3 \)

The intersection of these intervals gives us:

\[ -3 < a < 3 \]

### Conclusion

Thus, the values of the parameter \( a \) for which the function \( f(x) \) has points of extrema that are opposite in sign are:

\[ \boxed{(-3, 3)} \]