Answer:
The function f(x) = 2x³ + 3x² - 336x
is
Increasing on: (-∞, -8) U (7, ∞)
and
Decreasing on: (-8, 7)
Step-by-step explanation:
Given the function
f(x) = 2x³ + 3x² - 336x
We need to find the interval on which f is increasing, and the interval on which it is decreasing.
First, we take the derivative of f with respect to x to obtain
f'(x) = 6x² + 6x - 336
Next, set f'(x) = 0 and solve for x.
f'(x) = 6x² + 6x - 336 = 0
Dividing by 6, we have
x² + x - 56 = 0
(x - 7)(x + 8) = 0
x = 7 or x = -8
Now we have the interval
(-∞, -8) ∪ (-8, 7) ∪ (7, ∞)
Let us substitute a value from the interval (-∞, -8) into the derivative to determine if the function is increasing or decreasing.
Let us choose x = -9, then
f'(-9) = 6(-9)² + 6(-9) - 336
= 96 > 0
Since 96 is positive, the function is increasing on (-∞, -8).
Again, substitute a value from the interval (-8, 7) into the derivative to determine if the function is increasing or decreasing.
Choose -2
f'(-2) = 6(-2)² + 6(-2) - 336
= -324 < 0
Since -324 is negative, then the function is decreasing on (-8, 7)
Lastly, substitute a value from the interval (7, ∞) into the derivative
Choose x = 9
6(9)² + 6(9) - 336
= 204
The function is increasing on (8, ∞)
f'(x) > 0
Collating these, we have the function to be
Increasing on:
(-∞, -8) U (7, ∞)
Decreasing on:
(-8, 7)