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Mathematical Expression Editor
[?]
Throughout this module, if something does not exist, write DNE in the
answer box.
Recap Video
Take a look at the following video which recaps the ideas from the section.
Example Video
Below is a video showing a worked example.
Problems
There are a lot of theorems and definitions in this section, so let’s recall all of
them:
A critical point of a function is a point in the domain of where or
does not exist.
If has a local maximum or minimum at , then either or does not exist (i.e.
is a critical point of ).
If is a continuous function on the closed interval , then has an absolute
maximum and an absolute minimum on the interval .
Consider the following graph:
How many critical points does this function have? .
List the critical points in
increasing order:
, and this point on the graph is a
.
, and this point on the graph is a
.
, and this point on the graph is a
.
Consider the function . How many critical points does this function have? .
In increasing order, the critical points are:
Consider . Which of the following describes the full domain of ?
How many critical points does have? .
The critical point is at .
Here is the
graph of :
Based on this graph, the point is a local
.
Procedure 1. To find the absolute maximum and minimum values of a
continuous function on the closed interval :
Find the critical points of on (i.e. inside the interval)
Plug in these critical points and endpoints into to get the highest
and lowest -values.
Find the absolute maximum and minimum values of on the interval .
First, note that is continuous on the interval . Let’s break this into
steps:
Step 1: We can find critical points inside the interval. Taking a
derivative, we get
This derivative does not exist at , but that is ok in this case because
is one of the endpoints (and we only care right now about the
interval ). Setting , we get one critical point at , which is inside the
interval.
Step 2: We found is a critical point in the previous part. We also
have and as endpoints. We plug these three points into :
This tells us that the maximum value is at , and the minimum
value is at .
Find the absolute maximum and minimum values of the function on the
interval .
Find the absolute maximum and minimum values of the function on the
interval .
Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)
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Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)