Consider the equation below. f(x) = 2x3 + 3x2 − 336x (a) Find the interval on which f is increasing. (Enter your answer in interval notation.) (−[infinity],−8],[(7,[infinity])] Find the interval on which f is decreasing. (Enter your answer in interval notation.)

f is increasing = Solution Interval = x ∈ (-∞, -8] ∪ [7, ∞)

f is decreasing = Solution Interval = x ∈ [-8, 7]

Step-by-step explanation:

The given function is

f(x) = 2x³ + 3x² − 336x

We are required to find the interval on which f(x) is increasing and decreasing.

f(x) is increasing:

f(x) = 2x³ + 3x² − 336x

Taking the derivative of f(x) with respect to x yields,

f'(x) = 3*(2x²) + 2*(3x) - 336

f'(x) = 6x² + 6x) - 336

f'(x) ≥ 0

6x² + 6x - 336 ≥ 0

You can use either quadratic formula or use your calculator to solve the quadratic equation,

(x + 8) + (x - 7) ≥ 0

(x + 8) ≥ 0 or (x - 7) ≥ 0

x ≤ -8 or  x ≥ 7

so

x ∈ (-∞, -8]

x ∈ [7, ∞)

Solution Interval = x ∈ (-∞, -8] ∪ [7, ∞)

f(x) is decreasing:

f(x) = 2x³ + 3x² − 336x

We have already found the derivative f'(x)

f'(x) = 6x² + 6x - 336

f'(x) ≤ 0

6x² + 6x - 336 ≤ 0

(x - 7) + (x + 8) ≤ 0

(x - 7) ≤ 0 or (x + 8) ≤ 0

x ≤ 7 or x ≥ -8

-8 ≤ x ≤ 7

Solution Interval = x ∈ [-8, 7]

2

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