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