Q. 465.0( 1 Vote )

# The interval on w

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.

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