**Objectives:** In this tutorial, we investigate what derivatives tell us about local maxima and local minima. We apply these results to finding maxima and
minima of functions having only one critical points and functions which are continuous on a closed interval.
After working through these materials, the student should be able

- to find the local maxima and local minima of functions symbolically;
- to find the maximum and minimum of a function on a closed interval.

- Review the definitions of maximum, minimum, local maximum and local minimum.
**Theorem.**Suppose that**a**is a local maximum or a local minimum of the function**f**. Then either**f '(a) = 0**

**f '(a)**does not exist.

**Discussion**[Using Flash]**Examples.****f(x) = x**^{3}+ 3x^{2}- 24x + 3

Discussion.**f(x) = x**^{3}+ 5

Discussion.**f(x) = |x|**

Discussion.

**Definition.**Suppose that**a**is a point in the domain of the function**f**.**a**is a**critical point**of**f**if either**f '(a) = 0**or**f '(a)**does not exist.- The following theorem will be useful in later work.
**Theorem.**Suppose that the function**f**is differentiable on an interval**I**and suppose that the point**a**in**I**is the only critical point of**f**.- if
**a**is a local maximum of**f**then**a**is a maximum of**f**, and - if
**a**is a local minimum of**f**then**a**is a minimum of**f**.

- if
**Example.**- The maximum and minimum of the function
**f(x) = x**on the closed interval**[1, 5]**are the endpoints of the interval and are not critical points of**f**.

- The maximum and minimum of the function
- A very important property of continuous functions is the following theorem.
**Theorem.**Suppose that the function**f**is continuous on the closed interval**[a, b]**. Then- there exists
**c**in**[a, b]**such that**c**is a maximum of**f**, and - there exists
**d**in**[a, b]**such that**d**is a minimum of**f**.

**c**and**d**are either- critical points of
**f**or - endpoints of
**[a, b]**.

- there exists
**Examples.**- Find the maximum and minimum of a function of the form
**f(x) = a x**on a closed interval.^{2}+ b x + c

Discussion. - Find the maximum and minimum of a function of the form
**f(x) = a x**on a closed interval.^{3}+ b x^{2}+ c x + d

Discussion.

- Find the maximum and minimum of a function of the form
**Drill problems**on finding maxima and minima of functions on closed intervals.