Final answer:
The function f(x) = 1.66667x³ + 2.5x² - 30x is increasing on the open interval (-6, 2). This is found by first determining the critical points of the function by setting its derivative equal to zero, then checking the sign of f'(x) in the intervals separated by the critical points.
Step-by-step explanation:
To find the open intervals where the function f(x) = 1.66667x³ + 2.5x² - 30x is increasing, we need to first locate the critical points by setting the derivative equal to zero. The derivative of the function is f'(x) = 5x^2 + 5x - 30. By setting this equal to zero and factoring, we get (x - 2)(x + 6) = 0, thus the critical points are 2 and -6.
Next, we use a number line and test points in the intervals separated by the critical points. We choose -7 for (-∞, -6), -1 for (-6, 2), and 3 for (2, ∞). If f'(x) > 0 in a given interval, then f(x) is increasing in that interval.
f'(x) = 5x^2 + 5x - 30, so if we plug -7 into it, we get a negative number. That means f(x) is decreasing in (-∞, -6). For the interval (-6, 2), we plug in -1, and we get a positive number, and for (2, ∞), the test number 3 gives a negative number. So the function is increasing on the interval (-6, 2).
Learn more about Increasing Intervals