Explanation
Solving
1. Find the derivative f′(x)
The critical points (maxima and minima) occur where f′(x)=0.
f′(x)=dxd[2x3+(2p−7)x2+3(2p−9)x−6]
f′(x)=6x2+2(2p−7)x+3(2p−9)
2. Analyze the roots of the quadratic f′(x)=0
For f(x) to have a maxima at x < 0 and a minima at x > 0, the quadratic equation 6x2+2(2p−7)x+3(2p−9)=0 must have two distinct real roots of opposite signs (one negative, one positive).
3. Apply the condition for roots of opposite signs
For a quadratic equation ax2+bx+c=0 to have roots of opposite signs, the product of the roots must be negative:
\text{Product of roots} = \frac{c}{a} < 0
In our equation:
So, we require:
\frac{3(2p - 9)}{6} < 0
4. Final Set of Values
The values of p must be less than 29, which is represented as the interval:
Correct Option: (2)
