What is the First Derivative Test for Local Extreme Values? Give examples of how it is applied.

Step 1 of 3:

In this problem, we need to explain the First Derivative test for Local Extreme values with an example.

Step 2 of 3:

The First Derivative Test for Local Extreme Values:

First Derivative Test for Local Extrema is defined as if f has a local maximum or minimum value at interior point c of its domain then

If changes sign from negative to positive at c, then f has a local minimum at c.If changes sign from positive to negative at c, then f has a local maximum at c.

Now we discuss local maximum and local minimum.

Local Maximum:

According to the definition of The First Derivative Test for the Local Extreme value, a function f has a local maximum value at a point c within its domain D if for all lying in some open interval containing c.

Local Minimum:

According to the definition of The First Derivative Test for the Local Extreme value, a function f has a local minimum value at a point c within its domain D if for all lying in some open interval containing c.

Now we see by the graph: