Hence, pick a test value from each interval and plug it in to the first derivative. Minimums and maximums occur when the graph of f changes from decreasing to increasing or from increasing to decreasing. Theorem and the Mean Value Theorem is saying. Understand the hypotheses of these theorems and the counter examples that show why each of those hypotheses are important. Understand and be able to use the Increasing-Decreasing Test, the First Derivative Test and the Concavity Test.
Points of inflection occur where the graph of f changes concavity.
Using the same analysis as with the first derivative, you can determine where the graph of f is concave up and concave down. If the graph changes from concave up to concave down or visa-versa, then you have found a point of inflection. Be able to use the first and second derivatives of a function to obtain precise information about the graph of the function and be able to fill out the derivative table for the function you are trying to analyze and graph. Use limit techniques to determine asymptotes. And remember, in order to sketch an accurate graph, you must also plot a few points.