The open intervals on which f(x) is increasing are (-∝, 1) and (9, ∝) and the open intervals on which f(x) is decreasing is (1, 9)
The relative maxima is x = 1
How to determine the open intervals on which f(x) is increasing or decreasing
To do this, we find the critical points of the function and evaluate its derivative at each interval.
So, we have
f(x) = x³ - 15x² + 27x + 14
The derivative of f(x) is
f'(x) = 3x² - 30x + 27
Setting f'(x) = 0, we have
x = 1, 9.
Since f'(x) is a polynomial, it is defined for all real numbers.
So, our the points are x = 1, and 9
So, the increasing interval is (-∝, 1) and (9, ∝)
And the decreasing interval is (1, 9)
Based on the intervals where f(x) is increasing or decreasing, we can determine that:
f(x) has a relative maximum at x = 1 and f(x) has a relative minimum at x = 9