Question

Let f(x) = x3

- 6x2+9x + 18, then f(x) is strictly increasing in …

A) (-∞, 1) (B) [3,∞) (C) (-∞, 1) ∪[3,∞) (D) (1, 3)​

Answer 1

Answer:

f

′

(x)=3x

2

−12x+9=3(x

2

−4x+3)=3(x−3)(x−1).

If f(x) is decreasing, then

f

′

(x)<0⇒3(x−3)(x−1)<0

⇒(x−3)(x−1)<0⇒ either x−3>0,x−1<0

⇒x−3<0,x−1>0⇒ either x>3,x<1

⇒x<3,x>1⇒3<x<1 or 1<x<3

⇒x∈(1,3)[∵3<x<1 is not possible

Answer 2

Answer:

(D) (1,3).

Step-by-step explanation:

f ′ (x)=3x 2 −12x+9=3(x 2−4x+3)=3(x−3)(x−1).

If f(x) is decreasing, then

f ′ (x)<0⇒3(x−3)(x−1)<0

⇒(x−3)(x−1)<0⇒ either x−3>0,x−1<0

⇒x−3<0,x−1>0⇒ either x>3,x<1

⇒x<3,x>1⇒3<x<1 or 1<x<3

⇒x∈(1,3)[∵3<x<1 is not possible].

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