To determine in which intervals the function f(x) = x^3 - 3x^2 - 9x increases, we need to find the intervals for which the derivative of f(x) is positive.
Step 1: Finding the derivative
The derivative of a function gives us information about the gradient (slope) of the function at any given point. If the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing.
The derivative f'(x) can be found using Power Rule in differentiation, which states that the derivative of x^n, where n is any real number, is n*x^(n-1).
So the derivative of f(x) = x^3 - 3x^2 - 9x is:
f'(x) = 3x^2 - 6x - 9
Step 2: Checking when f'(x) is greater than 0
Now that we have the derivative, we need to check when f'(x) > 0. The function increases when f'(x) > 0 and decreases when f'(x) < 0.
Step 3: Analyzing each interval
1. For the interval (3, 16), f'(x) is positive, hence the function is increasing in the interval (3, 16).
2. For the interval (-infinity, -4), f'(x) is negative, hence the function is decreasing in this interval.
3. For the interval (1, 9), f'(x) is negative for some part of the interval and positive for the other part. Hence, the function is not only increasing in this interval.
4. For the interval (16, infinity), f'(x) is positive, hence the function is increasing in this interval.
5. For the interval (0, 3), f'(x) is negative, hence the function is decreasing in this interval.
6. For the interval (-4, 0), f'(x) is negative, hence the function is decreasing in this interval.
In summary, the function f(x) = x^3 - 3x^2 - 9x only increases in the interval (3, 16).