What Derivatives Tell Us and
Applications to Graphing

Theorem
Maximum
(a,f(a))
Minimum
(b,f(b))
This maximum and minimum must occur at a critical value or at an end point.

Example

Solution

A Slight Change to the Previous Problem

Another Example

The Rest of the Solution

What if the interval is not closed or bounded?
What if f(x) is discontinuous?
If this is the case, then f(x) need not have a maximum or a minimum.

Look at this function defined on [2, 8]
This function does not have a maximum.
The maximum is not 10.
It has a minimum which is 3.

Here’s another example where the minimum or maximum may not exist.

The Answers
The minimum value of f(x) on [0,1) is 1.
There is no maximum value.

There is no maximum value.
The minimum exists and is -2.

If a function defined on an open interval has a maximum or minimum value, then these extrema must occur at a critical value.

An example of a function defined from – infinity to +infinity
This function f(x) is defined for all real numbers.
It has a maximum value, but it is does not have a minimum.
Note that the maximum occurs at a critical value.

Finding the Maximum Value on an
Unbounded Interval

Conclusion of Example

Tangent line lies above curve

Look at the slopes of the tangent line when the curve is concave up.
If the slope is increasing then y’ is increasing which means y” > 0 .

Theorem about Concavity
If y” > 0 on an open interval (a,b) then the curve is concave upward on this interval.
If y”< 0 on an open interval (a,b) then the curve is concave downward on this interval.

The Rest of the Solution

Here is the graph

The Rest of the Problem

The Graph

Example

The Solution
Make test values but be careful

The relative extrema occur when x = 1 and x = -1.
There are 3 inflection pts. They occur when x = -.707, 0, and .707