1. Open Box Problem
Problem: What is the maximum volume of an open box that can be created by cutting out the corners of a 20 cm x 20 cm piece of cardboard?

2. Let’s look at the graph

3. Graph of the derivative

4. Extrema
Let the function f be defined on an interval containing the value c.
f(c) is the minimum of f on the interval if 𝑓(𝑐)≤𝑓 𝑥 for all x in the interval.
f(c) is the maximum of f on the interval if 𝑓(𝑐)≥𝑓 𝑥 for all x in the interval.
The maximum and minimum of a function on an interval are the extreme values. They are also called the absolute minimum and absolute maximum on the interval.

5. Extreme Value Theorem
If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval.

6. Relative Extrema
Any “hill” is a relative maximum, any “valley” is a relative minimum.
1) If there is an open interval containing c which f(c) is a maximum, then f(c) is called a relative maximum.
2) If there is an open interval containing c on which f(c) is a minimum then f(c) is called a called a relative minimum.

7. Critical Numbers
Let f defined at c. If f’(c) = 0 or if f is not differentiable at c, then c is a critical number of f.
If f has a relative minimum or relative maximum at x = c, then c is a critical number of f.

8. Finding Extrema on a Closed Interval
To find the extrema of a continous function f on a closed interval [a,b], use the following steps.
1) Find the critical numbers of f in (a,b).
2) Evaluate f at each critical number
3) Evaluate f at each endpoint of [a,b].
4) The least of these values is a minimum. The greatest is the maximum.

9. Examples
Find the extrema of 𝑓 𝑥 =3 𝑥 4 −4 𝑥 3 on the interval [-1, 2].
Find the extrema of 𝑓 𝑥 =2𝑥−3 𝑥 2/3 on the interval [-1, 3].
Find the extrema of 𝑓 𝑥 =2 sin 𝑥 − cos 2𝑥 on the interval [0, 2𝜋].

10. Hw pg. 209
Pg. 13,15,17,21,23,29,33,37