**Objectives:** In this tutorial, we discuss the use of the derivative to
detect whether a function is increasing or decreasing. In addition, we define
the maximum, minimum, local maximum and local minimum of a function.
After working through these materials, the student should be able

- to recognize graphically when a function is increasing or decreasing by looking at the graph of its derivative;
- to recognize graphically the local maximum and the local minimum of a function.

**Modules:**

- Recall the following definition.
**Definition.**A function**f**is**increasing on the interval I**if, for each**a < b**in**I**,**f(a) < f(b)**. A function**f**is**decreasing on the interval I**if, for each**a < b**in**I**,**f(a) > f(b)**. - Intuitively, by looking at the graphs of increasing and decreasing functions, the following theorem
appears to be reasonable.
**Theorem.**If the derivative of a function**f**is positive for all**x**in an interval**I**, then**f**is increasing on**I**.If the derivative of a function

**f**is negative for all**x**in an interval**I**, then**f**is decreasing on**I**. - The proof of this Theorem will be given in another module.
**Example.****f(x) = x**^{3}+ 8 x^{2}- 3 x + 5

Discussion.

**Using a graphing calculator**to graph a function and its derivative.- TI-86 Graphing Calculator [Using Flash]

**Definition.**Suppose that**a**is in the domain of the function**f**such that, for all**x**in the domain of**f**,**f(x)**__<__f(a)then

**a**is called**a maximum**of**f**.Suppose that

**a**is in the domain of the function**f**such that, for all**x**in the domain of**f**,**f(x)**__>__f(a)then

**a**is called**a minimum**of**f**.- Finding maxima and minima is one of the major problems which calculus helps us solve.
**Definition.**Suppose that**a**is in the domain of the function**f**and suppose that there is an open interval**I**containing**a**which is contained in the domain of**f**such that, for all**x**in**I**,**f(x)**__<__f(a)then

**a**is called**a local maximum**of**f**.Suppose that

**a**is in the domain of the function**f**and suppose that there is an open interval**I**containing**a**which is also contained in the domain of**f**such that, for all**x**in**I**,**f(x)**__>__f(a)then

**a**is called**a local minimum**of**f**. **Discussion**[Using Flash]