UPSC NDA Math Model Paper 3

Given:
f(x)=−2x3−9x2−12x+1
Differentiating, both sides, we get
⇒f’(x)=−6x2−18x−12
For f(x) to be decreasing function,
∴f’(x)<0
⇒−6x2−18x−12<0
⇒−6(x2+3x+2)<0
Multiplying each side of an inequality by a negative number reverses the direction of the inequality symbol
⇒(x2+3x+2)>0
⇒(x+2)(x+1)>0 ∴x∈(−∞,−2)∪(−1,∞)
∴ Option 4 is correct.