Certainly, to find the intervals where a function is increasing involves calculus. Here are the steps:
1. **Define the Function**: This typically would be given in the problem. Let's arbitrarily use f(x) = x^3 as an example for this walk-through.
2. **Find the Derivative**: The derivative of a function gives us the rate of change of that function at any given point. For f(x) = x^3, the derivative would be f'(x) = 3x^2.
3. **Set the Derivative Equal to Zero**: This will help us to find critical points, which are places where the function might change from increasing to decreasing, or vice versa. Setting the derivative equal to zero gives us 3x^2 = 0.
4. **Solve for x**: The values of x we get here are known as critical points as they represent the points where the rate of change of the function could change from positive (increasing) to negative (decreasing), or vice versa. Solving for x yields x = 0.
5. **Test Intervals**: Now that we know our critical points, we need to test those intervals on the derivative (NOT the original function) to see where the function is increasing or decreasing. We take test points that fall within each interval and plug them into our derivative.
Split intervals are: x < 0 and x > 0. We can take any point that falls within these intervals. Let's take test points -1 for the interval (x < 0) and +1 for the interval (x > 0). Substituting these values in the derivative we get f'(-1) = -3 and f'(1) = 3. The negative result for the interval x < 0 indicates a decreasing function, whereas the positive result for the interval x > 0 indicates an increasing function.
6. **State the Interval**: Based on test results, we conclude that the function increases for the interval x > 0 (which in interval notation is (0, infinity)).
And that's it! We've found where the function f(x) = x^3 is increasing. The function is increasing in the interval (0, infinity).