To find out the intervals in which the function f(x) = -(1/3)(x+6)² + 12 is increasing, we'll follow these steps:
1. Find the derivative (f') of the function. The derivative of a function gives us the slope of the function at any given point. So for a function to be increasing, the derivative should be greater than zero (f'(x) > 0).
2. The derivative of f(x) = -(1/3)(x+6)² + 12 using the power rule (which says that the derivative of x^n, where n is any real number, is n*x^(n-1)) is f'(x) = -2/3*(x+6).
3. Set the derivative greater than zero and solve for x.
-2/3*(x+6) > 0
Rearranging, we get:
x < -6
So, the function f(x) = -(1/3)(x+6)² + 12 is increasing where x is in the interval (-∞, -6).
Note: The intervals are usually written in the form (a, b) where 'a' is the starting point, 'b' is the end point, and the parentheses mean the end points are not included in the interval. In this case, the interval (-∞, -6) means x can be any number less than -6, and -∞ is just a way to represent the smallest possible number.