Wetzel                                                                                                            

Math 140

3.1 – Extrema on an Interval

Review –        

                        Graph the function,           (open interval)

            Absolute Max. & Min. – (Extreme Values or Extrema)

            Relative Max. & Min. –   (Relative Extrema)

Intervals –      Now take only the piece of the function on the closed interval [-1, 5]

            What are the max and mins now?

            The open interval (-1, 5)?

The Extreme Value Theorem –

            If  is continuous on a closed interval , then  has both a minimum and a maximum on the interval.

Critical Numbers –

            def. – the x-value of a point where the derivative of a function is zero or does not exist.

Ex.:  Find all critical numbers of f(x).

1.)                                               2.) 

Finding Extrema on a Closed Interval –

            1.)  Find the critical numbers of f(x) on the interval (a, b).

            2.)  Evaluate f(x) at each critical number in (a, b).

            3.)  Evaluate f(x) at each endpoint of [a, b].

            4.)  The greatest of these values is the maximum.  The least is the minimum.

Ex.:  Find the absolute extrema.

3.)  ,  [-1, 1]                                  4.)  ,  [-2, 4]

HW:  p.160  #13, 15, 19-29odd, 37, 39