How do you find the turning points of a cubic function?

The basic way to find turning points (maximums and minimums) of any function is the first derivative test. You to take the derivative of the function and set it equal to zero (##f'(x) = 0##) or find where it does not exist. Those values are called critical numbers and are the only places the function can change direction.

Once you have your critical values you check to see how the derivative changes sign at those points. To do this you put in an x-value on both sides of your critical point into the derivative. (For example if your critical point is x = 2 then see what the sign is of ##f'(1)## and of ##f'(3)##) If the derivative changes from Positive (increasing) to Negative (decreasing) then you have a maximum. If the derivative changes from Negative to Positive you have a minimum and if it has the same sign on both sides the graph does not change direction there.

You can find a good video about this at Khan Academy:Testing Critical Points for local extrema