1. Minimum and Maximum Values of Functions
4.6
Minimum and Maximum Values of Functions
Borax Mine, Boron, CA
Photo by Vickie Kelly, 2004
Greg Kelly, Hanford High School, Richland, Washington

2. Extreme values can be in the interior or the end points of a function.
No Absolute
Maximum
Absolute Minimum

3. Absolute
Maximum
Absolute Minimum

4. Absolute
Maximum
No Minimum

5. No
Maximum
No Minimum

6. Extreme Value Theorem:
If f is continuous over a closed interval, then f has a maximum and minimum value over that interval.
Maximum & minimum
at interior points
Maximum & minimum
at endpoints
Maximum at interior point, minimum at endpoint

7. Finding Maximums and Minimums Analytically:1
Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points. 2
Find the value of the function at each critical point. 3
For closed intervals, check the end points as well.
For open intervals, check the limits as x approaches the end points to determine if min or max exist within interval.

8. EXAMPLE FINDING ABSOLUTE EXTREMA
Find the absolute maximum and minimum values of
on the interval
There are no values of x that will make
the first derivative equal to zero.
The first derivative is undefined at x=0,
so (0,0) is a critical point.
Because the function is defined over a
closed interval, we also must check the
endpoints.

9. At:
At:

10. At:
At:
At:
Absolute
minimum:
maximum:

11. EXAMPLE FINDING ABSOLUTE EXTREMA
Find the absolute maximum and minimum values of
on interval
Critical point at x = 1.5.
Because the function is defined over a
open interval, we also must check the
limits at the endpoints.

13. Absolute
minimum:
maximum: