Find the intervals in which the function f(x) = -2x³ – 9x² – 12x + 1 is

(i) Strictly increasing

(ii) Strictly decreasing

f(x) = -2x³ – 9x² – 12x + 1

Now to find the intervals in which f(x) increases and decreases we first have to find the critical points

Critical points are those points or values of x at which f’(x) = 0

Hence differentiate y with respect to x and equate it to 0

⇒ 0 = -6x² – 18x – 12

⇒ x² + 3x + 2 = 0

⇒ x² + 2x + x + 2 = 0

⇒ x(x + 2) + 1(x + 2) = 0

⇒ (x + 1)(x + 2) = 0

⇒ x = -1 and x = -2

Now we have to check for intervals (-∞, -2), (-2, -1) and (-1, ∞)

Case1:

When x < -2 that is x in (-∞, -2)

Observe that (x + 1) and (x + 2) are negative for x < -2 hence their product (x + 1)(x + 2) is positive. But we also have a -3 hence -3(x + 1)(x + 2) will be negative

Hence hence f(x) is decreasing

Case 2:

When -2 < x < -1 that is x in (-2, -1)

Observe that (x + 1) is negative when x is in between -2 and -1 and (x + 2) is positive when x is in (-2 , -1) hence their product (x + 1)(x + 2) is negative. But we have -3 multiplied to it hence the product -3(x + 1)(x + 2) will be positive

Hence hence f(x) is increasing

Case 3:

When x > -1 that is x in (-1, ∞)

Observe that (x + 1) and (x + 2) are positive for x > -1 hence their product (x + 1)(x + 2) is positive. But we also have a -3 hence -3(x + 1)(x + 2) will be negative

Hence hence f(x) is decreasing

Hence the conclusion of case1, case2 and case3 is that f(x) is increasing in the interval (-2, -1) and decreasing in the interval (-∞, -2) U (-1, ∞)