Final answer:
The given function is a polynomial function. To find the intervals on which the function is increasing or decreasing, we need to find its derivative and determine the sign of the derivative in different intervals. By finding the critical points and testing intervals, we can determine that the function is decreasing on the intervals (-∞, -5/3) and (-5/3, -1), and increasing on the interval (-1, ∞).
Step-by-step explanation:
The function y = 3ˣ³ + 12ˣ² + 15x is a polynomial function. To determine the intervals on which the function is increasing or decreasing, we need to find the derivative of the function and determine where it is positive or negative. Let's find the derivative of the function:
y' = 9ˣ² + 24ˣ + 15
Setting the derivative equal to zero and solving for x, we find the critical points:
9ˣ² + 24ˣ + 15 = 0
(3ˣ + 3)(3ˣ + 5) = 0
x = -1 or x = -5/3
Now, we can test the intervals between these critical points to determine where the function is increasing or decreasing. Let's choose test points:
-2, -1, -4/3, 0, 1
Substituting these test points into the derivative, we find the signs:
x = -2: y'(-2) = 9(-2)² + 24(-2) + 15 = 39, positive
x = -1: y'(-1) = 9(-1)² + 24(-1) + 15 = 0, zero
x = -4/3: y'(-4/3) = 9(-4/3)² + 24(-4/3) + 15 = -1, negative
x = 0: y'(0) = 9(0)² + 24(0) + 15 = 15, positive
x = 1: y'(1) = 9(1)² + 24(1) + 15 = 48, positive
From these signs, we can determine the intervals:
Decreasing: (-∞, -5/3) and (-5/3, -1)
Increasing: (-1, ∞)
Therefore, the function is decreasing on the intervals (-∞, -5/3) and (-5/3, -1), and increasing on the interval (-1, ∞).