Math, asked by khusiram210820, 5 months ago

find the intervals in which the function f(x)=2x^3 -15x^2-36x+6 is increasing or decreasing​

Answers

Answered by TheValkyrie
20

Answer:

Step-by-step explanation:

Given:

f(x) = 2x³ - 15x² - 36x + 6

To Find:

The intervals in which the function is increasing or decreasing

Solution:

f(x) = 2x³ - 15x² - 36x + 6

First differentiating the given function with respect to x,

f'(x) = 6x² - 30x - 36

Taking 6 as common,

f'(x) = 6 (x² - 5x - 6)

Factorizing by splitting the middle term,

f'(x) = 6 (x² - 6x + 1x - 6)

f'(x) = 6 (x - 6) (x + 1)

Putting f'(x) = 0 we get,

6 (x - 6) (x + 1) = 0

x = 6, x = -1

This divides the number line into 3 disjoint intervals,

( -∞, -1) and  (-1, 6) and (6, ∞)

Checking if f'(x) > 0 or f'(x) < 0 in the intervals

In the interval (-∞ , -1)

At x = -2,

6 ( -2 - 6) (-2 + 1) > 0

f'(x) > 0

Hence in the interval (-∞, -1), the function is strictly increasing.

Now in the interval (-1, 6)

At x = 0,

6 ( 0 - 6) (0 + 1) < 0

f'(x) < 0

Hence in the interval (-1, 6) the function is strictly decreasing.

In the interval (6, ∞)

At x = 10,

6 ( 10 - 6) (10 + 1) > 0

f'(x) > 0

Hence in the interval (6, ∞) the function is strictly increasing.

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