Answer:
decreasing: (-∞, -9) ∪ (8, ∞)
increasing: (-9, 8)
Explanation:
You want the intervals where the function f(x) = -2x³ -3x² +432x +1 is increasing and decreasing.
Derivative
The slope of the graph is given by its derivative:
f'(x) = -6x² -6x +432 = -6(x +1/2)² +433.5
Critical points
The slope is zero where ...
-6(x +1/2)² = -433.5
(x +1/2)² = 72.25
x +1/2 = ±8 1/2
x = -9, +8
Intervals
The graph will be decreasing for x < -9 and x > 8, since the leading coefficient is negative. It will be increasing between those values:
decreasing: (-∞, -9) ∪ (8, ∞)
increasing: (-9, 8)
__
Additional comment
A cubic (or any odd-degree) function with a positive leading coefficient generally increases over its domain, with a possible flat spot or interval of decrease. When the leading coefficient is negative, the function is mostly decreasing, with a possible interval of increase, as here.
<95141404393>