First+Derivative+Test

=__The First Derivative Test__=

We have seen so far how the derivative can be used to determine the slope of tangent lines to the graphs of curves at certain points. Now we will broaden the use of the derivative to analyze to behavior of curves over intervals.

The critical points on a curve are the points at which the behavior of the graph changes. These occur at the points where the first derivative equals zero or is undefined.
 * //Finding the Critical Points//**

Find the critical points of the graph f(x)=2x 2 +3x+1.
 * Example One **

Solution: We will first take the derivative: Below, you will see a graph of the equation from Example 1. You should be able to see that it does change behavior, in this case changing from decreasing to increasing, at x=(-3/4). (This graph obtained through www.coolmath.com/graphit)

We can now use the critical points with the First Derivative Test to predict the behaviors of curves without graphing them.
 * //The First Derivative Test//**

The rules for the First Derivative Test are as follows: For a function that is continuous and differentiable at a critical point x=a 1. If f '(x)>0 on an interval extending from the left of a, and f '(x)<0 on an interval extending to the right of a, then f(a) is a local maximum of f(x). 2. If f '(x)<0 on an interval extending from the left of a, and f '(x)>0 on an interval extending from the right of a, then f(a) is a local minimum of f(x). 3. If f '(x) has the same sign on intervals to both sides of a, then f(a) is an inflection point of f(x).

Use the first derivative test to describe the behavior of the graph: f(x)=2x 2 +3x+1.
 * Example Two **

Solution: In Example 1, we found that f '(x)=4x+3, and that there is a critical point at x=(-3/4). Test points to either side of the critical point to find the graph's behavior. f '(-1)=4(-1)+3=(-1), f '(0)=4(0)+3=3 As f '(x)<0 to the left, and f '(x)>0 to the right, there is a **relative minimum at x=(-3/4)**. In other words, f(x) is Decreasing on x<(-3/4) and is Increasing on x>(-3/4), which is confirmed by the graph in Example One.