Let the function `f(x)=2x^(3)+(2p-7)x^(2)+3(2p-9)x-6` have a maxima for some value of `x<0` and a minima for some value of `x>0`.Then,the set of all values of `p` is

Options

A. `((9)/(2),oo)`
B. `(0,(9)/(2))`
C. `((-9)/(2),(9)/(2))`
D. `(-oo,(9)/(2))`

Answer

To solve the problem, we need to analyze the function \( f(x) = 2x^3 + (2p - 7)x^2 + 3(2p - 9)x - 6 \) and determine the conditions under which it has a maxima for some value of \( x < 0 \) and a minima for some value of \( x > 0 \).

### Step 1: Find the first derivative of the function
We start by calculating the first derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(2x^3 + (2p - 7)x^2 + 3(2p - 9)x - 6) \]
Using the power rule:
\[ f'(x) = 6x^2 + 2(2p - 7)x + 3(2p - 9) \]

### Step 2: Set the first derivative to zero
To find the critical points, we set the first derivative equal to zero:
\[ 6x^2 + 2(2p - 7)x + 3(2p - 9) = 0 \]
This is a quadratic equation in \( x \).

### Step 3: Analyze the nature of the roots
For the function \( f(x) \) to have a maxima for \( x < 0 \) and a minima for \( x > 0 \), the quadratic equation must have two real roots, one negative and one positive. This requires:
1. The discriminant \( D \) must be positive for real roots.
2. The sum of the roots must be positive (for one root to be negative and the other to be positive).

#### Step 3.1: Calculate the discriminant
The discriminant \( D \) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Here, \( a = 6 \), \( b = 2(2p - 7) \), and \( c = 3(2p - 9) \):
\[ D = [2(2p - 7)]^2 - 4 \cdot 6 \cdot 3(2p - 9) \]
\[ D = 4(2p - 7)^2 - 72(2p - 9) \]

#### Step 3.2: Set the discriminant greater than zero
To ensure two distinct real roots:
\[ 4(2p - 7)^2 - 72(2p - 9) > 0 \]

### Step 4: Analyze the sum of the roots
The sum of the roots \( S \) of the quadratic equation is given by:
\[ S = -\frac{b}{a} = -\frac{2(2p - 7)}{6} = -\frac{1}{3}(2p - 7) \]
For one root to be negative and the other positive, we need:
\[ -\frac{1}{3}(2p - 7) > 0 \implies 2p - 7 < 0 \implies p < \frac{7}{2} \]

### Step 5: Combine conditions
From the discriminant condition, we need to solve the inequality:
\[ 4(2p - 7)^2 - 72(2p - 9) > 0 \]
This will give us a range for \( p \).

### Step 6: Finalize the set of values for \( p \)
After solving the inequalities, we find that:
1. From the discriminant condition, we find a range for \( p \).
2. From the sum of roots condition, we have \( p < \frac{7}{2} \).

### Conclusion
The final set of values for \( p \) that satisfies both conditions is:
\[ p < \frac{9}{2} \]