Question

The interval on which the function f (x) = 2x³ + 9x² + 12x – 1 is decreasing is:

A. [–1, ∞)
B. [–2, –1]
C. (–∞, –2]
D. [–1, 1]

Answer 1

Given f (x) = 2x³ + 9x² + 12x – 1

Applying the first derivative we ge

Applying the sum rule of differentiation and also the derivative of the constant is 0, so we get

Applying the power rule we get

⇒ f’(x)=6x²+18x+12-0

⇒ f’(x)=6(x²+3x+2)

By splitting the middle term, we get

⇒ f’(x)=6(x²+2x+x+2)

⇒ f’(x)=6(x(x+2)+1(x+2))

⇒ f’(x)=6((x+2) (x+1))

Now f’(x)=0 gives us

x=-1, -2

These points divide the real number line into three intervals

(-∞, -2), [-2,-1] and (-1,∞)

(i) in the interval (-∞, -2), f’(x)>0

∴ f(x) is increasing in (-∞,-2)

(ii) in the interval [-2,-1], f’(x)≤0

∴ f(x) is decreasing in [-2, -1]

(iii) in the interval (-1, ∞), f’(x)>0

∴ f(x) is increasing in (-1, ∞)

Hence the interval on which the function f (x) = 2x³ + 9x² + 12x – 1 is decreasing is [-2, -1].

So the correct option is option B.