**OKAAAY!**Well, once again, I can't upload my graphs/pics. So I'll go early tomorrow morning at school, as usual, and complete this post, or go on again later and see if it works. Sorry for the inconvenience.

Today’s class was about

**Global Extrema**and the

**Extreme Value Theorem**.

**TERMS**

**Maxima**– plural for maximum

**Minima**– plural for minimum

**Extreme**– either a minimum or a maximum

**Extrema**– minima and maxima

**Local**– just a vicinity on a graph

**Global**– considers the entire interval

**Critical Numbers**– where f’ = 0 or undefined

In previous lessons, we learned that

*local extrema*are the lowest or highest possible points in

*some vicinity on a graph*. For example, given this graph (figure A), we’ll see that the points highlighted with green appear to be the maximum and minimum. If we consider the entire graph, however, we know that these points aren’t the extrema.

Global extrema are the maxima and minima over the entire interval. Notice that I said “over the entire interval”.

**The interval must be set to determine the global extrema**.

To find the global extrema, do the following steps:

1) Find f’

2) Find the critical numbers

3) Evaluate f at the critical numbers

**and**

**end points****

4) Determine the global extrema (the lowest and highest values on the interval)

**Always check where the function is undefined. This is the most overlooked place where the function has a min/max

This leads to a theorem in calculus called the

**Extreme Value Theorem**(

**EVT**), which states that if a function is continuous on a closed interval, [a, b], then it has a global max and global min. This is an

**existence theorem**, meaning that if you can draw the graph of a function without lifting your pen (continuous), then global extrema must exist. It doesn’t say

*where*these exist; just says they

*should*exist.

Let’s do an example, following the steps above:

What we can take away from this is that from the interval [-1, 3], the global extrema are

**also**the local extrema.

Due for Wednesday are Exercises 5.1, odds.

Also, due on Monday is the derivative photo assignment.

## 2 comments:

great post christian! i was just wondering who the next scribe is?!? anyway, good luck in getting those pictures/graphs up.

WHOA! THANKS MR. K!!!! THANKS LOTS -_-

I tried to upload it again today, but the error keeps on coming up.

THAAAANKS MR. K!!!

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