Write down the function of interest to begin the problem. This will most likely be referenced from your textbook. For this example, f(x) = 2x^3 + 4x^2 + 2x + 29.
Take the first derivative f'(x) of your function. Using the usual rules of differentiation, you get f'(x) = 6x^2 + 8x + 2.
Set f'(x) equal to zero and factor the resultant polynomial that represents the first derivative. This will show you where the first derivative of the function is equal to zero, and therefore which points represent potential extrema. For our example, you have f'(x) = 6x^2 + 8x + 2 = 0 = (6x + 2)(x + 1). The zeros of this equation are x = -1/3 and x = -1.
Use the zeros determined in Step 3 as the end bounds of the intervals you will be testing. These shall be written as (-infinity, -1), (-1, -1/3) and (-1/3, infinity) for the example.
Evaluate the first derivative of a test point from each interval. This will tell you how the function is behaving in each interval, allowing you to determine if the extremum is a minimum or a maximum. For the interval (-infinity, -1), look at f'(-2) = 6(-2)^2 + 8(-2) + 2 = 10 > 0. When f'(x) > 0, the function is increasing (and when f'(x) < 0, the function is decreasing). For the interval (-1, -1/3) we look at f'(-1/2) = 6(-1/2)^2 + 8(-1/2) + 2 = -1/2 < 0. Since f'(x) is decreasing on the right-hand side of the point x = -1 and increasing on the left-hand side, we conclude that x = -1 is a maximum. For the interval (-1/3, infinity), we look at f'(1) = 6(1)^2 + 8(1) + 2 = 14 > 0. For the point x = -1/3, f(x) is decreasing on the left-hand side and increasing to the right indicating we now have a minimum.