You can use sign charts to analyze the behavior of a function. They help you find maxima, minima and saddle points. Here, it’s important to keep your head in the game. You’re looking to say something about the function based on its derivative . This is how you do it:
Rule
Find the derivative of . Draw the sign chart of the differentiated function . You’re going to find out where this function is above and below the -axis.
Mark which -values give you positive -values (solid line) and which ones give negative -values (dashed line).
Rule
It turns out that there is a clear connection between and :
When is positive (above the -axis), increases.
When is negative (below the -axis), decreases.
When is zero (on the -axis), has a maximum, minimum or saddle point.
Rule
Example 1
You have a cubic function . Find the maxima and minima of .
where and are the solutions of . That means the factorized expression for the derivative is
Now you need to find the corresponding -values. To do so, just insert the -values you found into the function
That gives you:
Minimum point:
because
because
Between the minimum and the maximum the graph increases, because the derivative of the function is positive and its sign chart has a solid line.
From the maximum and onward the graph decreases again, because the derivative of the function is negative and its sign chart has a dashed line.