1. Section 4.1 Maximum and Minimum Values
AP Calculus
October 20, 2009
Berkley High School, D1B1

2. Max and Min Of course they can’t be too easy; this IS calculus.
Absolute Maximum: f has an absolute maximum if there exists a c such that f(c)≥f(x), for x in D. f(c) is called the maximum value.
Absolute Minimum: f has an absolute minimum if there exists a c such that f(c)≤f(x), for x in D. f(c) is called the minimum value.
The absolute max and absolute min are called the extremes.
Calculus, Section 4.1

3. Max and Min
Local Maximum: f has a local maximum if there exists a c such that f(c)≥f(x), for all x near c.
How close to c? Calculus close.
Local Minimum: f has a local minimum if there exists a c such that f(c)≤f(x), for all x near c.
Endpoints of closed intervals can not be local maximums or minimums.
Calculus, Section 4.1

4. Example Local Minimum and Absolute Minimum Local Maximum
Local Maximum and Absolute Maximum
Local Minimum
Calculus, Section 4.1

5. Local and Absolute Maximum
Example
Local Minimum
Local and Absolute Maximum
Local Maximum
Local Minimums
Absolute Minimum
Calculus, Section 4.1

6. Local and Absolute Maximum
Example
Local Maximum
Local and Absolute Maximum
Absolute Minimum
Local Minimum
Local Minimum
Calculus, Section 4.1

7. Critical Numbers
Definition: “A critical number of a function f is a number c in the domain of f such that either f’(c) = 0 or f’(c) does not exist.”
Theorem: “If f has a local maximum or minimum at c, then c is a critical number of f.”
Translation: Any critical number has the potential of being a local maximum of minimum. Only at critical numbers can a local max or a local min exist.
Calculus, Section 4.1

8. x=0 is a critical value, but not a local maximum or minimum
Example
x=0 is a critical value, but not a local maximum or minimum
Calculus, Section 4.1

9. Example
f(0)=0 is a local minimum. Because the derivative at 0 is undefined, 0 is a critical value.
Calculus, Section 4.1

10. Closed Interval Method
To find the absolute maximum and minimum values of a continuous function f on a closed interval [a, b]:
Find the values of f at the critical numbers of f on (a,b)
Find the values of f at the endpoints of the interval.
The largest of the values for steps 1 & 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.
Calculus, Section 4.1

11. Example
Calculus, Section 4.1

12. Example
Absolute Minimum
Absolute Maximum
Calculus, Section 4.1

13. Assignment
Section 4.1, 15-55, odd
Calculus, Section 4.1